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Consider the following optimization problem:
\[ \begin{align} \max ~~ & \sum_{i=1}^{n} f(x_i) \\ \text{subject to~~} & \sum_{i=1}^{n} x_{i} = a \\ & x_{i} \in \{0,\dots,c_i\} \forall i \in \{1,\dots,n\}. \end{align} \]
Here, \(f(x)\) is a non-linear function of \(x\) and is defined for each non-negative integer \(x\). ... |
I am at a loss how to transform the following function:
$\qquad \Bigg[\Bigg(\omega^{2}-\alpha+\beta*z^{4}\Bigg)^{2}+ (\delta*\omega)^{2}\Bigg]*z^{2} = \gamma$
into an explicit dependance of $z$ on $\omega$, i.e., $z = f(\omega)$. I have tried solving the equation using Matlab and Mathematica, but both programs return o... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
5.1.1 Boundary value problems
We have encountered several different eigenvalue problems such as:
\[ X''(x)+ \lambda X(x)=0\]
with different boundary conditions
\(X(0)=0~~~~~X(L)=0~~~~~{\rm{(Dirichlet)~or,}}\)
\(X'(0)=0~~~~~X'(L)=0~~~~~{\rm{(Neumann)~or,}}\)
\(X'(0)=0~~~~~X(L)=0~~~~~{\rm{(Mixed)~or,}}\)
\(X(0)=0~~~~~X'(L)... |
The following script generates two plots in a row, both of them controlled by Manipulate. I would like to combine them together in one.
The problem I don't know how to solve is that the plotting region in
Q is different for the two plots, even though it is included in the range $Q \in [0,1]$.
Manipulate[Row[{ Plot3D[{ ... |
I am reading Huber's Robust Statistics (2nd). On page 2 and 3 he gave an example. The basic facts are summarized here. Let $(X_n)$ be a sequence of random variables and define two measures of spread as follows.
Mean Absolute Deviation: $d_n := \frac{1}{n}\sum|x_i-\bar x|$. Standard Deviation: $s_n := \sqrt{\frac{1}{n}\... |
Covers of Sets in a Topological Space
Definition: Let $X$ be a topological space and let $A \subseteq X$. Then a Cover/Covering of $A$ is a collection of subsets $\mathcal F$ of $X$ such that $\displaystyle{A \subseteq \bigcup_{U \in \mathcal F} U}$. If $\mathcal F$ is a collection of open sets that satisfies the above... |
Example Problems Regarding Equivalent Binary Quadratic Forms
Recall from the Equivalent Binary Quadratic Forms page that if $f$ and $g$ are binary quadratic forms then $f$ is said to be equivalent to $g$ if there exists $m_{11}, m_{12}, m_{21}, m_{22} \in \mathbb{Z}$ such that:(1)
and $m_{11}m_{22} - m_{12}m_{21} = 1$.... |
T4 Normal Hausdorff Topological Spaces
Recall from the T0 Kolmogorov Topological Spaces page that a topological space $X$ is said to be a T
0 space or a Kolmogorov space if for every pair of distinct points $x, y \in X$, $x \neq y$ there exists open neighbourhoods $U$ of $x$ and $V$ of $y$ such that either $x \not \in ... |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane... |
Attention Conservation Notice:A tedious exercise in elementary probability theory showing that \(R^{2}\) doesn’t behave the way you think. But you already knew that.
It is a fact universally known that, when the simple linear regression model \[ Y = \beta_{0} + \beta_{1} X + \epsilon\] is correct, the value of \(R^{2}\... |
It is the half-twist. If you have an $2n$-gon, then rotate it by $n$. This is your centre (along with the trivial element).
Note that for $G=\langle a, b, \ldots, c\rangle$ we have that $g\in Z(G)\Leftrightarrow ga=ag, gb=bg, \ldots, gc=cg$. That is, $g$ is in the centre if and only if it commutes with all the generato... |
I am looking for a solution to a wave equation
$\frac{\partial^2 u}{\partial \tau^2} = \frac{\partial^2 u}{\partial \xi^2}$
in which $t_c\tau = t$, $L\xi = x$,
and $t_c = L/v_c$ is the characteristic time,
$L$ is the sample thickness,
and $v_c$ is the characteristic wave speed,
with an IC of
$\left [\frac{\partial u}{\... |
Difference between revisions of "Literature on Carbon Nanotube Research"
(→Ultra-high-yield growth of vertical single-walled carbon nanotubes: Hidden roles of hydrogen and oxygen)
Line 86: Line 86:
industry.
industry.
−
== Ultra-high-yield growth of vertical single-walled carbon nanotubes
+
== Ultra-high-yield growth o... |
Newform invariants
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
\(\beta_{0}\) \(=\) \( 1 \) ... |
I believe this integral $$\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$$
can not be computed exactly. However is there a method or transformation to express this integral in terms of the cosine integral or similar? I am referring to the integrals here.
$a$ is real number; with the change of variable this integral... |
I recently saw the identity $$ \frac{1}{1} - {1 \over 2} +{1 \over 3} - {1 \over 4} + {1 \over 5} - {1 \over 6} \dotsb = \log(2) $$ which I found rather interesting. I was intrigued by the way a transcendental number like $\log(2)$ could be expressed by using the reciprocals of every integer once. I then wondered what ... |
The Method of Integrating Factors
Let $p$ and $g$ be functions of $t$ and consider the following first order differential equation:(1)
If we multiply the both sides of the equation above by the function $\mu (t)$ we get that:(2)
If we can guarantee that $\mu ' (t) = \mu (t) p(t)$, then notice that by applying the produ... |
Four balls are transfered from an urn holding 3 white and 5 black balls. Two balls are drawn from the new urn and are white and are
then replaced; which I presume means they were first drawn without replacement.
Let random variable $X$ count the number of white balls transfered to the new urn, $E$ the event that two wh... |
I am trying to solve the following problem:
$$\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x), \ x > 0, t > 0$$
$$\phi(x, 0) = 0, \ x > 0$$
$$\phi(0, t) = e^{-t}, \ t > 0$$
Taking the Laplace transform (in $t$, of course) gives
$$\mathcal{L} \left\{ \dfrac{\partial{\phi}}{\partia... |
Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. With a double integral we can handle two dimensions and variable density.
Just as before, the coordinates of the center of mass are
$$\bar x={M_y\ove... |
In Exercises \((2.4E.1)\) to \((2.4E.14)\), find a particular solution.
Exercise \(\PageIndex{1}\)
\(y''-3y'+2y=e^{3x}(1+x)\)
Answer
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{2}\)
\(y''-6y'+5y=e^{-3x}(35-8x)\)
Answer
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{3}\)
\(... |
I'm reading through some notes one locally convex spaces ("lcs" from now on) analysis and there the following version of the Banach-Steinhaus theorem is given
Theorem (Banach-Steinhaus)$\quad$ The pointwise limit of a sequence of continuous, linear mappings from a barrelled lcs $U$ to a lcs $V$ is again a continuous, l... |
Here I transfered the question from the comment The relationship between spin and spinor curvature
How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + \frac{1}{4} \epsilon_{\mu\nu}\gamma^\mu\gamma^\nu \Psi$ arise?
When $\mathcal{R}_{ab}$ is defined by
$\newcommand{\Rcal}{\mathcal{... |
Joint work with Øystein Linnebo, University of Oslo.
J. D. Hamkins and Ø. Linnebo, “The modal logic of set-theoretic potentialism and the potentialist maximality principles,” to appear in Review of Symbolic Logic, 2018.
@ARTICLE{HamkinsLinnebo:Modal-logic-of-set-theoretic-potentialism, author = {Hamkins, Joel David and... |
Connectedness of Finite Topological Products
Recall that if $X$ and $Y$ are topological spaces and $X \times Y$ is the Cartesian product of $X$ and $Y$, then we defined the product topology on $X \times Y$ to be the topology whose basis is:(1)
We define the topological product of $X$ and $Y$ to then be $X \times Y$ wit... |
When $k=2$ and $n$ is even,$$D_{k,r} S^n \simeq BB_{r+} \wedge S^{rn}$$is the $rn$-fold unreduced suspension of the classifying space of the braid group $B_r$ on $r$ strings. I once cited Cohen-Mahowald-Milgram ``The stable decomposition for the double loop space of a sphere'' (1978) for this, and if I recall correctly... |
Root-Finding Techniques for Polynomials
We will now look at a few techniques to find roots of polynomials. The first theorem will tell us that if we have a rational root (in lowest terms) to a polynomial with integer coefficients, then the numerator/denominator of this rational root is restricted to be certain values.
... |
I am trying to understand whether some non trivial explicit estimate is known about the following $$\sum_{n \leq x} \mu(n) \chi(n)$$ as a function of $x$.
Here $\chi$ is a Dirichlet character modulo $q$ and $q > (\log x)^2$ (for example).
MathOverflow is a question and answer site for professional mathematicians. It on... |
For a Hohmann transfer to Saturn, I get 15.7 km/s for both burns. The transfer time is also a simple formula. I obtain roughly
6 years.
Compare to the lunar ice. It is roughly
2.8 km/s to get to, and the trip time would be a few days, even from Low Earth Orbit.
As suggested by the other answer, you could compare to Ear... |
The X-15 had a reaction control system for all three axes using thrusters with hydrogen-peroxide monopropellant. There was an automatic as well as a manual mode. The manual mode used a single three-axis control joystick.There were two completely independent systems. Each system used six RCS thrusters, two for each axis... |
"The concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible." A. Kolmogorov
For continuous random variables, $X$ and $Y$ say, conditional distributions are defined by the property that they recover the original probability measure, that is, for all measurab... |
In my post Trigonometry Yoga, I discussed how defining sine and cosine as lengths of segments in a unit circle helps develop intuition for these functions.
I learned the circle definitions of sine and cosine in my junior year of high school, in the class that would now be called pre-calculus (it was called “Trig Senior... |
This question already has an answer here:
Why is E(XY)=E(XE(Y|X))?
Is this using the properties of conditional expectation and is there a general formula that can be applied when you have E(...)=E(..(E(Y|X))?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analy... |
I'm implementing a 2D physics engine where collision response is based on impulse computation. I'm going to first expose the context, then the problem, and finally the questions.
Context
Simple collision between two solids, the first one is static, the second one falls. The second solid has its colliding face parallel ... |
κ(G) ≤ λ(G) ≤ δ(G) ≤ 2|E(G)|/|V(G)| ≤ Δ(G)
Relation of Min Vertex Cutsets, Min Edge Cutsets, The Minimum Degree, and Maximum Degree
We are going to prove the following inequality relating the minimum vertex cutset $\kappa (G)$, the minimum edge cutset $\lambda (G)$, the minimum degree in a graph $\delta (G)$, and the m... |
The Cauchy-Schwarz Inequality
We are about to look at one of the more significant inequalities in mathematics known as the Cauchy-Schwarz inequality. Before we do so though, we will first need to be able to write each vector $u$ in an inner product space as a sum of a scalar multiple of a vector $v$ and a vector $w$ th... |
amp-mathml
Displays a MathML formula.
Required Script
<script async custom-element="amp-mathml" src="https://cdn.ampproject.org/v0/amp-mathml-0.1.js"></script>
Supported Layouts container Examples amp-mathml.amp.html
This extension creates an iframe and renders a MathML formula.
<amp-mathml layout="container" data-form... |
My question is the following:
Suppose $M$ is an $n \times n$ symmetric real matrix. I want to find an $n \times n$ symmetric real matrix X such that $|| X -M||_F$ is minimized with the constraint that $U^T X U \succeq 0$ (positive semidefinite), where $U$ is an $n \times d$ matrix, $U^TU = I_d$ and $n > d$. $|| \cdot |... |
We can use prime factorization to find the smallest common multiple of two positive integers.
The least common multiple (l.c.m.) of two positive integers is the smallest positive integer that is a multiple of both.
We denote the least common multiple of two positive integers \(a\) an \(b\) by \(\langle a,b\rangle\).
\(... |
The plane through the ray and the origin intersects your ellipsoid to form a planar ellipse. To find this planar ellipse one must establish a basis vector on the plane such that any point on the plane can be represented by two parameters ($\alpha$ and $\beta$). $$ \vec{r} = \alpha \vec{p} + \beta \vec{d} $$
The ellipse... |
This is a technical problem which must have been solved already. It won't be in beginners textbooks but there should be a solution somewhere. I welcome reading suggestions. Maybe someone with experience in solving Navier-Stokes equations numerically can help me. Here goes:
An incompressible fluid flowing down a pipe ob... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message zedler Joined: 03 Mar 2006 Posts: 15
Posted: Mon Mar 27, 2006 11:53 am Post subject: spacing of mathrm Hello,
\documentclass{book}
\usepackage{times,mtpro2}
\begin{document}
$(\mathrm j$
\end{document}
gives touching glyphs.
Michael Michael Spi... |
The Norm of a Vector
Definition: If $\vec{u} \in \mathbb{R}^n$, then the Norm or Magnitude of $\vec{u}$ denoted $\| \vec{u} \|$ is defined as the length or magnitude of the vector and can be calculated using the formula: $\| \vec{u} \| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2}$.
We will note that the norm of a vector is som... |
Here’s a simple and intuitive way of looking at the geometry of a least squares regression:
Take the bottom left point in the triangle below as the origin O. For the linear model:
$$ Y=X\beta + \epsilon $$
Both $Y$ and \(X\beta\) are vectors, and the residual vector \(\epsilon\) is the difference. The standard least sq... |
General Background
In this class we dealt with various crystals models, like the tableau models. However, many of these models only work for specific classes of Kac-Moody Lie algebras (eg finite dimensional), and often must be changed to fit with any particular Lie algebra. One of the ideas behind the model presented i... |
The Set of Accumulation Points under Homeomorphisms on Topological Spaces
Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open.
Furthermore, if such a homeomorphism exists then w... |
Answer
$-\frac{2}{5}$
Work Step by Step
1.Reciprocal identity $\cos\theta=\frac{1}{\sec\theta}$ 2. Substitute the given value: $\cos\theta=\frac{1}{-\frac{5}{2}}=-\frac{2}{5}$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hours to send in... |
Euler's Method for Approximating Solutions to Differential Equations
We are now going to look at a numerical method for approximating a solution of a first order differential equation initial value problem known as
Euler's Method or The Tangent Line Method.
Suppose that $\frac{dy}{dt} = f(t, y)$ is a first order differ... |
It's impossible to find explicit formulas for solutions of some differential equations. Even if there are such formulas, they may be so complicated that they're useless. In this case we may resort to graphical or numerical methods to get some idea of how the solutions of the given equation behave.
In Section~3.4 we'll ... |
Covering Spaces
Definition: Let $X$ be a topological space. A Covering Space of $X$ is a pair $(\tilde{X}, p)$ where: 1. $\tilde{X}$ is a path connected and locally path connected topological space. 2. $p : \tilde{X} \to X$ is a continuous surjective mapping such that for every $x \in X$ there is an path connected open... |
I have solution that can guarantee the end goal being reached in a maximum of
four five steps. (Many thanks for @dmg and @Taemyr for their comments to fix this solution.) The trick is to:
Reduce the cups into the following configuration:
$\begin{bmatrix} \circ & \bullet \\ \bullet & \circ \end{bmatrix} $
where turning ... |
I can't seem to figure out how to do the following: compute the implied correlation $ρ_{imp}$ by using the observed market price $M_{quote}$ of a Margrabe option, and solving the non-linear equation shown below:
$$M_{quote} = e^{−q_0T}\times S_0(0)\times N(d_+)−e^{−q_1T}\times S_1(0)\times N(d_−)$$
where:
$$\begin{alig... |
GolfScript (23 chars)
{:^((1${\.**2^?%}+*}:f;
The sentinel result for a non-existent inverse is
0.
This is a simple application of Euler's theorem. \$x^{\varphi(2^n)} \equiv 1 \pmod {2^n}\$, so \$x^{-1} \equiv x^{2^{n-1}-1} \pmod {2^n}\$
Unfortunately that's rather too big an exponential to compute directly, so we have... |
Focus Questions
The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. What is a trigonometric equation? What does it mean to solv... |
I try to give an answer, welcome corrections!
When we write a state, we must notice the reference frame which the state lies in, because the form of state is diffrent in diffrent frame. Now I give 3 frames $O,O', O''$. Their correlations are $t'=t-\tau'$, $t''=t-\tau''$, with $\tau'=-\infty$, $\tau''=+\infty$. Let's sp... |
Difference between revisions of "Capillary waves"
(Stuff added from my contribution to wikipedia -- still to be rewritten a bit)
m (Tidy up (still work to do))
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===Derivation===
===Derivation===
−
This is a sketch of the derivation of the general dispersion relation, see Ref.
+
This is a sketch of the ... |
2019-10-09 06:01
HiRadMat: A facility beyond the realms of materials testing / Harden, Fiona (CERN) ; Bouvard, Aymeric (CERN) ; Charitonidis, Nikolaos (CERN) ; Kadi, Yacine (CERN)/HiRadMat experiments and facility support teams The ever-expanding requirements of high-power targets and accelerator equipment has highligh... |
I have a question where I need to find the general solution of the differential equation $y-3y^{7}=(y^{3}+6x)y'$, where the solution is in the form $F(x,y)=C$. I am only concerned about finding $F(x,y)$, and I have been advised to rewrite this equation in differential form, but I am still unsure where to go from here. ... |
You can do it by combining grammars.
Notation:
$A,B,C,...$ will be non-terminals
$a,b,c,...$ will be terminals
$\alpha,\beta, ...$ will be "whatever can be here"
$x$ and $y$ will be regular expressions
The grammar corresponding to a regular expression $x$ will be denoted $f(x)=(N_x,\Sigma,P_x,S_x)$
$f(a)=(\{S\},\Sigma,... |
Connected and Disconnected Topological Spaces
One very important characteristic of a topological space is the concept of connectedness and disconnectedness. We define these terms below.
Definition: Let $X$ be a topological space. Then $X$ is said to be Disconnected if there exists open sets $A, B \subset X$ such that $... |
The queen wishes to build a patio paved with of a circular center stone surrounded by circular rings of circular stones. All the stones in a ring will be the same size with the same number of stones in each ring. The stones in the innermost ring will be placed touching (tangent to) the adjacent stones in the ring and t... |
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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 ... |
Sometimes it is useful to compute the length of a curve in space; for example, if the curve represents the path of a moving object, the length of the curve between two points may be the distance traveled by the object between two times.
Recall that if the curve is given by the vector function \(\bf r\) then the vector ... |
I can provide an approach by finite elements and an application of the functional calculus of selfadjoint operators.
Background
The spectrum of the Laplacian on a bounded domain $\varOmega$ with sufficiently smooth boundary subject to homogeneous Dirichlet boundary conditions is discrete. By the spectral theorem, there... |
Search
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 ... |
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 $A\subseteq\kappa$... |
x[1] := 1
x[n_] := n - 1 + Sum[j x[j], {j, 1, n - 1}]
x /@ Range[10]
{1, 2, 7, 29, 146, 877, 6140, 49121, 442090, 4420901}
Edit
$$x(1)=1$$
$$x(n+1)=n+\sum\limits_{j=1}^n j\cdot x(j) = 1+n-1+\sum\limits_{j=1}^{n-1}j\cdot x(j)+n\cdot x(n) = 1+x(n)+n\cdot x(n) \Rightarrow$$
$$x(n+1) = 1+(n+1)\cdot x(n)$$
but then
$$x(2)=x... |
In practice, sound is a complex multitude of waves with different frequencies, phase, and amplitudes. When sound is recorded digitally, however, it becomes simply a collection of numbers. The properties of the many sound waves – frequency, phase, amplitude – cannot be discerned from the digitized data. In the world of ... |
Difference between revisions of "Vopenka"
(bigger separate section)
Line 7: Line 7:
For any language $\mathcal{L}$ and any proper class $C$ of $\mathcal{L}$-structures, there are distinct structures $M, N\in C$ and an [[elementary embedding]] $j:M\to N$.
For any language $\mathcal{L}$ and any proper class $C$ of $\math... |
Search
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 ... |
Charged-Lepton-Flavour Violation in the light of the Super-Kamiokande data 42 Downloads Citations Abstract.
Motivated by the data from Super-Kamiokande and elsewhere indicating oscillations of atmospheric and solar neutrinos, we study charged-lepton-flavour violation, in particular the radiative decays \(\mu \rightarro... |
I have a data set where many of the actual values are zero, so I can't use MAPE. It's not a time series, so I can't use MASE ala our very own Rob Hyndman.
Is there another alternative to MAPE that I could use?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data anal... |
Modelling Population Growth and Decay
One important application of differential equations arises in modelling population growth or decay within a species. Let $y = y(t)$ denote the population size of a species at time $t$. If the rate of change of this population is proportional by some constant $r$ (where $r > 0$ impl... |
Tabulating numbers
Let $u(n,k)$ denote the number of upwards-pointing triangles of size $k$ included in a triangle of size $n$, where size is a short term for edge length. Let $d(n,k)$ likewise denote the number of down triangles. You can tabulate a few numbers to get a feeling for these. In the following table, row $n... |
Some programming principles help us make short work of this. The key principle is to encapsulate what's going on.
First, the surface. It depends on some parameters, so let's be explicit about that, rather than letting those parameters run around loose as "global" variables. To illustrate, I begin by generating some (re... |
In this paper the authors perform a high-temperature expansion of the correlation functions for a Heisenberg model on a lattice. Starting from
$$\left<\mathbf{S}_i\cdot\mathbf{S}_j\right>_\beta = \mathrm{Tr}(\mathbf{S}_i\cdot\mathbf{S}_j\,e^{-\beta H})/\mathrm{Tr}\, (e^{-\beta H})$$
They write
expanding in powers of $\... |
The PhD thesis of Paul J. Werbos at Harvard in 1974 described backpropagation as a method of teaching feed-forward artificial neural networks (ANNs). In the words of Wikipedia, it lead to a "rennaisance" in the ANN research in 1980s.
As we will see later, it is an extremely straightforward technique, yet most of the tu... |
▶▶ ◀◀
▶
Announcements
▶
Schedule
Room A208. Tue and Thu, 9-11.
Teacher's office hours: Mon 9-11..
▶
Course description Goal:
Introduction to complex analysis.
Contents:
Complex numbers and geometry of the complex plane.
Analytic functions, contour integrals and power series.
Cauchy theory and applications.
▶
Books and ... |
J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
@ARTICLE{HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD, author = {Joel David Hamkins and Jonas Reitz}, title = {The se... |
The Hilbert Basis Theorem
Recall from the Noetherian Rings page that a ring $R$ is said to be a Noetherian ring if it satisfies the ascending chain condition, that is, for all ascending chains of ideals $I_1 \subseteq I_2 \subseteq ... \subseteq I_n \subseteq ...$ there exists an $N \in \mathbb{N}$ such that for all $m... |
Difference between revisions of "Literature on Carbon Nanotube Research"
m (→Tensile and Electrical Properties of Carbon Nanotube Yarns and Knitted Tubes in Pure or Composite Form)
Line 16: Line 16:
* Moderator: [[mailto:Markus.Landgraf@me.com Markus Landgraf]]<br />
* Moderator: [[mailto:Markus.Landgraf@me.com Markus ... |
The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) injective model structure and has a mapping space functor $Hom$, given by $Hom(X, Y)_n = hom(X\times \Delta^n, Y)$.
Using t... |
For a sequence $X_1, X_2, \dots $, Let $F_n(x)$ denote the cdf of $X_n$.
Suppose our sequence is $X_n \sim N(0,n) $ then for all $x$ the point-wise limit of $F_n(x)$ is $\frac{1}{2}$.
How would one prove this?
Cross Validated is a question and answer site for people interested in statistics, machine learning, data anal... |
Solution for $J_1$But what is J1? What is the expected number of steps for a frog that has only one leaf to go?The solution is $J_1 = 2(e-1)$ and other terms $J_n$ can be expressed as a sum.Rewriting the recurrence relation as a sumThe recurrence relation is not gonna solve the problem entirely (because one term in the... |
Huge cardinal Huge cardinals (and their variants) were introduced by Kenneth Kunen in 1972 as a very large cardinal axiom. Kenneth Kunen first used them to prove that the consistency of the existence of a huge cardinal implies the consistency of $\text{ZFC}$+"there is a $\omega_2$-saturated $\sigma$-ideal on $\omega_1$... |
Compact T2 Spaces are T4 Spaces
Recall from the T2 (Hausdorff) Topological Spaces page that a topological space $X$ is said to be T
2 or Hausdorff if for all $x, y \in X$, $x \neq y$ there exists open neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U \cap V = \emptyset$.
Also recall from the T4 (Normal Hausdorff) T... |
Here’s a simple and intuitive way of looking at the geometry of a least squares regression:
Take the bottom left point in the triangle below as the origin O. For the linear model:
$$ Y=X\beta + \epsilon $$
Both $Y$ and \(X\beta\) are vectors, and the residual vector \(\epsilon\) is the difference. The standard least sq... |
2019-07-18 17:03
Precision measurement of the $\Lambda_c^+$, $\Xi_c^+$ and $\Xi_c^0$ baryon lifetimes / LHCb Collaboration We report measurements of the lifetimes of the $\Lambda_c^+$, $\Xi_c^+$ and $\Xi_c^0$ charm baryons using proton-proton collision data at center-of-mass energies of 7 and 8 TeV, corresponding to an... |
I have an unweighted, undirected graph $G$ with $(v_1, ..., v_N) \in V$ vertices, where each vertex in the lattice represents a string of some length ($\leq 200$ characters), and I need to construct $G$ from an unordered list of edges between vertices (i.e. a list saying that vertex $v_i$ is connected to vertex $v_j$ f... |
The set $V= \{ A\in M_n(\mathbb{C})\ |\ A\bar A = I\}$ is a smooth submanifold of $M_n(\mathbb{C})$ with real dimension $n^2$.
Proof: Consider the involution $\iota:\mathrm{GL}(n,\mathbb{C})\to \mathrm{GL}(n,\mathbb{C})$ defined by$$\iota(A) = (\bar A)^{-1}.$$This is an anti-holomorphic involution of the complex manifo... |
What is the difference between Base Correlation and Implied Correlation for a CDO tranche?
An implied correlation $\rho_i(k_1,k_2)$ is a correlation that matches the $(k_1,k_2)$ tranche price $P_{k_1}^{k_2}$ (usually computed under a gaussian or student t copula)
$$ C(k_1,k_2,\rho_i(k_1,k_2)) = P_{k_1}^{k_2} $$
For mez... |
This post was first published on 12/29/15, and has since been migrated to Blogger.
This is a tutorial on implementing Ian Goodfellow's Generative Adversarial Nets paper in TensorFlow. Adversarial Nets are a fun little Deep Learning exercise that can be done in ~80 lines of Python code, and exposes you (the reader) to a... |
101 people flip a fair coin. Everyone who tosses heads is on one team and everyone who tosses tails is on another other team. The team with more people on it wins. What are the odds that, given you are one of the 101 players, you will win? (101 players and coins eliminates ties but I am also interested the case where t... |
Model
A
model of a theory $T$ is a set $M$ together with relations (eg. two: $a$ and $b$) satisfying all axioms of the theory $T$. Symbolically $\langle M, a, b \rangle \models T$. According to the Gödel completeness theorem, in $\mathrm{PA}$ (Peano arithmetic) (so also in $\mathrm{ZFC}$) a theory has models iff it is ... |
Table of Contents
Repeated Roots of The Characteristic Equation Examples 1
Recall from the Repeated Roots of The Characteristic Equation page that for a first order linear homogenous differential equation with constant coefficients, $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$ where $a, b, c \in \mathbb{R}$, then ... |
Finishing some long overdue work, I was recently benchmarking different ways to compute orbital closures of permutation groups.
Orbital Closures, and what others call them
Let’s first lay down what I am computing. I have referred to Orbital Graphs before.
Suppose that \Omega = \{1..n\} for some n \in \mathbb{N} and G \... |
This sum may be evaluated by considering the following contour integral in the complex plane:
$$\oint_C dz \frac{\pi \cot{\pi z}}{(m z^2-n)^2}$$
where $C$ is a rectangular contour that encompasses the poles of the integrand in the complex plane, up to $z=\pm \left ( N +\frac12\right)$, where we consider the limit as $N... |
V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem.” (manuscript under review)
@ARTICLE{GitmanHamkinsKaragila:KM-set-theory-does-not-prove-the-class-Fodor-theorem, author = {Victoria Gitman and Joel David Hamkins and Asaf Karagila}, title = {Kelley-Morse set theor... |
The set \(\mathbb{Z}\) of all integers, which this book is all about, consists of all positive and negative integers as well as 0. Thus \(\mathbb{Z}\) is the set given by \[\mathbb{Z}=\{...,-4,-3,-2,-1,0,1,2,3,4,...\}.\] While the set of all
positive integers, denoted by \(\mathbb{N}\), is defined by \[\mathbb{N}=\{1,2... |
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