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Difference between revisions of "Image Dimensions"
(Added a bit of description of the properties dialog)
m
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;The on-screen size(?): The fields ''Width'' and ''Height'' tell synfigstudio how many pixels the image shall cover at a zoom level of 100%.
;The on-screen size(?): The fields ''Width'' and ''Hei... |
Thanks all for participating POW actively. Here’s the list of winners:
1st prize: Lee, Myeongjae (이명재) – 2012학번 2nd prize: Kim, Taeho (김태호) – 수리과학과 2011학번 3rd prize: Park, Minjae (박민재) – 2011학번 4th prize: Suh, Gee Won (서기원) – 수리과학과 2009학번 5th prize: Lim, Hyunjin (임현진) – 물리학과 2010학번
Congratulations! We again have very g... |
Ab Initio “A journey of a thousand miles begins with a single step.” –Lao Tzu.
When someone is studying competitive programming from the beginning, he or she is said to be a student of competitive programming
ab initio. Students should be given relatively easy problems to begin their journeys, before they are gradually... |
The Gell-Mann matrices $\lambda^\alpha$ are the generators of $SU(3)$.
Applying an SU(3) - transformation on the triple $q = ( u , d, s )$ of 4-spinors looks like this:
$$ q \rightarrow q' = e^{i \Phi_\alpha \lambda^\alpha / 2} q.$$
So far I can follow and I also understand why the expression $\bar{q}q$ is invariant un... |
783 1
I was told that the general definition of a derivative is
[tex]f'(x) = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}[/tex] (supposed to be delta y over delta x, but I can't make the latex work )
but why can't it work when [itex]\Delta y \rightarrow 0[/itex]?
[tex]f'(x) = \lim_{\Delta x \rightarrow 0} \f... |
In geometry,
parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lin... |
The eigenstates of a time-independent Hamiltonian$$ H |\psi_j \rangle = E_j|\psi_j\rangle$$ have the usual rotating-phase time dependence in the Schrödinger picture:$$ |\psi_j(t)\rangle = |\psi_j(t_0)\rangle \cdot \exp(E_j(t-t_0)/i\hbar) $$However, your formula indicates that $H_0$ and $V$ are time-dependent. So if the... |
The implication here is that each individual discoverer must start from nothing but a bag of crying cells, and build up knowledge in a linear order before making a discovery in a vacuum.
In reality, I find we have an entire interwoven society trying to make the discoveries, not independent individuals. There is an enti... |
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. ... |
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. ... |
Subset is Right Compatible with Ordinal Exponentiation Theorem
Let $x, y, z$ be ordinals.
Then: $x \le y \implies x^z \le y^z$ Proof
The proof shall proceed by Transfinite Induction on $z$.
Basis for the Induction
If $z = \varnothing$, then $x^z = 1$
\(\displaystyle x^z\) \(=\) \(\displaystyle 1\) Definition of Ordinal... |
RGPV First Year Engineering (Set A) (Semester 1)
Engineering Mathematics -I December 2011
Engineering Mathematics -I
December 2011
Total marks: --
Total time: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons (2) Marks are given to the right of every question (3) Draw neat diagrams where... |
As @EricWofsey pointed out, your construction doesn't work because you're taking a possibly infinite disjunction, but first-order logic only has finite disjunctions.
Here's one way to go about solving the problem:
First, we can assume without loss of generality that $\varphi$ has no constant symbols. (If it has constan... |
Let $s$ be a real number greater than 1. Let the function $f$ be defined as
$f(x)=\frac{ln(x)}{x^s}, x>0$.
a) Show that $\int_{a}^{\infty}\frac{ln(x)}{x^s}dx$ converges for $a>0$ and find its value.
b) Show that the infinite series $\sum_{n=1}^{\infty} \frac{ln(x)}{n^s}$ converges.
I believe I can show that (a) converg... |
Efficiently estimating recall June 09, 2019 tl;dr Factorize recall measurement into a cheaper precision measurement problem and profit. Measuring precision tells you where your model made a mistakebut measuring recall tells you where your model can improve. Estimating precisiondirectly is relatively easy but estimating... |
ZFITTER
ZFITTER v. 6. 21: A semi-analytical program for fermion pair production in e + e - annihilation. Characteristics of fermions pair production is important for the study of the properties of the Z-boson and for precision tests of the Standard Model at linear colliders at higher energies. High precision correction... |
If two topological spaces are weak homotopy equivalent to each other, are their Cech cohomology groups the same?
$$T = \left\{ \left( x, \sin \frac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,y)\mid y\in[-1,1]\}$$
This has trivial homotopy groups in degrees $\ge1$ but according to Wikipedia nontrivial Čech cohomolo... |
Let $T$ be the time-ordering operator which orders operators $A_1(t_1), A_2(t_2), \ldots$ such that the time parameter decreases from left to right:
$$T[A_1(t_1) A_2(t_2)] = A_2(t_2) A_1(t_1) \text{ if } t_2 > t_1 \text{ and }= A_1(t_1)A_2(t_2) \text{ otherwise. } $$
The time $t_i$ does not have to be a physical time, ... |
In my lecture notes I have that the distribution of a random variable $Y$ is said to be in the exponential family if it can be written as $f(y;\theta)=exp(a(y)b(\theta)+c(\theta)+d(y))$, where $a,b,c$ and $d$ are fixed functions. If we have (in the exponential family form) that $a(y)=y$, then the distribution is said t... |
In Appendix C of a paper by Michael E. Tipping and Christopher M. Bishop about mixture models for probabilistic PCA, the probability of a single data vector $\mathbf{t}$ is expressed as a mixture of PCA models (equation 69):
$$ p(\mathbf{t}) = \sum_{i=1}^M\pi_i p(\mathbf{t}|i) $$
where $\pi$ is the mixing proportion an... |
Data on the mean multiplicity of strange hadrons produced in minimum bias proton--proton and central nucleus--nucleus collisions at momenta between 2.8 and 400 GeV/c per nucleon have been compiled. The multiplicities for nucleon--nucleon interactions were constructed. The ratios of strange particle multiplicity to part... |
We all know that the density of the nucleus is very high.
Nuclei are made up of protons and neutrons, and while protons have the same charge, they are closely packed in a nucleus. How does the repulsion between protons not break apart the nucleus?
Chemistry Stack Exchange is a question and answer site for scientists, a... |
We have seen that sometimes double integrals are simplified by doing them in polar coordinates; not surprisingly, triple integrals are sometimes simpler in cylindrical coordinates or spherical coordinates. To set up integrals in polar coordinates, we had to understand the shape and area of a typical small region into w... |
My friend and I were building paper cones made from circles in which we cut out sectors of the circle and joined the two sides together. We were wondering about the relationship between the angle of the sector of the circle we cut out and the inclined angle of the cone the circle eventually made. While doing the math, ... |
Let $X$ be a smooth manifold, perhaps oriented if necessary. The frame bundle $\pi:P \to X$ carries a canonical trivialization of the pullback of the tangent bundle of $X$ and thus a canonical trivialization of each Stiefel-Whitney class of $TX$. I want to describe these geometrically.
The goal is to describe a $w_n$ s... |
How to Implement the Fourier Transformation from Computed Solutions
We previously learned how to calculate the Fourier transform of a rectangular aperture in a Fraunhofer diffraction model in the COMSOL Multiphysics® software. In that example, the aperture was given as an analytical function. The procedure is a bit dif... |
Defining parameters
Level: \( N \) = \( 33 = 3 \cdot 11 \) Weight: \( k \) = \( 2 \) Nonzero newspaces: \( 4 \) Newforms: \( 5 \) Sturm bound: \(160\) Trace bound: \(2\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(33))\).
Total New Old Modular forms 60 39 21 Cusp forms 2... |
M5: Multivariable Calculus - Material for the year 2019-2020
16 lectures
In these lectures, students will be introduced to multi-dimensional vector calculus. They will be shown how to evaluate volume, surface and line integrals in three dimensions and how they are related via the Divergence Theorem and Stokes' Theorem ... |
Per OP's wish, here's the math.SE answer I link to in my comment above.
Maybe it's worthwhile to talk through where the dual comes from on an example problem. This will take a while, but hopefully the dual won't seem so mysterious when we're done.
Suppose with have a primal problem as follows.
$$ Primal =\begin{Bmatrix... |
You can use
ListConvolve to simulate a single diffusion time step and build a simulation out of that. I'll show a simple example: Let's say we start with simple initial conditions like in your example
(initialconditions = Normal@SparseArray[{{3, 3} -> 1}, {5, 5}]) // MatrixForm
and a diffusion kernel
kernel = {
{1/120,... |
We start with a simulated Poisson example where the \(\Theta_i\) are drawn from a chi-squared density with 10 degrees of freedom and the \(X_i|\Theta_i\) are Poisson with expectation \(\Theta_i:\)
\[ \Theta_i \sim \chi^2_{10} \mbox{ and } X_i|\Theta_i \sim \mbox{Poisson}(\Theta_i) \]
The \(\Theta_i\) for this setting, ... |
I hope that the title isn't too provocative. ;-)
Bill Z. has brought my attention to a December 2012 nuclear physics paper that was updated 3 days ago,
The detailed calculations are concerned with the Hoyle state. What is it and what did the authors of the new paper conclude?
In 1954, Fred Hoyle noticed that we were lu... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
Difference between revisions of "LaTeX:Symbols"
(→Operators)
(→See Also)
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This article will provide a short list of commonly used LaTeX symbols.
This article will provide a short list of commonly used LaTeX symbols.
− − − − − − − − −
== Finding Other Symbo... |
According to the BHK interpretation of intuitionistic logic we have that:
A proof of $\exists x \in A . \phi(x)$ consists of a pair $(a, p)$ where $a \in A$ and $p$ is a proof of $\phi(a)$. A proof of $\forall x \in A . \phi(x)$ is a method which takes as input $a \in A$ and outputs a proof of $\phi(a)$. A proof of $\p... |
Background: I am a theoretical computer scientist (PhD candidate) and have done graduate level courses in Algebra.
I want to understand the following theorem from the book "Symmetric Bilinear Forms" by J. Milnor and D. Husemoller.
Theorem (9.5, pp 46) For any dimension n there exists a positive definite inner product s... |
Regression (OLS) - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the diff... |
This nice problem was in the analysis section of Putnam and Beyond: prove
\begin{align*}
\lim_{n\to \infty} n^2 \int_0^{1/n} x^{x+1} \, dx = 1/2. \end{align*}
The solution is quite nice, and simply relies on the fact that $\lim_{n\to 0^+} x^x = 1$, hence for $n$ large enough, we can approximate the integral with $\int_... |
Let $P_{n}$ be the product of the numbers in row of Pascal's Triangle. Then evaluate $$ \lim_{n\rightarrow \infty} \dfrac{P_{n-1}\cdot P_{n+1}}{P_{n}^{2}}$$
closed as off-topic by heropup, TZakrevskiy, user147263, dustin, Newb Mar 11 '15 at 0:19
This question appears to be off-topic. The users who voted to close gave t... |
I'm reading Boyd's
Convex Optimization textbook. In particular, I'm currently focusing on Chapter 5 (Duality). There is a frequent recurrence of two examples: Minimum volume covering ellipsoid\begin{align*} minimize & \quad log \; det \; X^{-1}\\ subject \; to & \quad a_i^T X a_i \leq 1 \\ \end{align*} Entropy Maximiza... |
I would like help with the conjecture that the function $f:\mathbb{R}_{\ge 0}^n\to\mathbb{R}_{\ge 0}^n $ with $f(x) = x \circ (Ax)$ where
∘ is the Hadamard product (equivalently, $f_i(x) = x_i \sum_j a_{ij}x_j$ )
and where A is a symmetric real matrix with elements $a_{ij} \ge 1$,
is invertible (locally would be enough... |
Finding equilibrium points of a continuous-time model \(\frac{dx}{dt} = G(x)\) can be done in the same way as for a discrete-time model, i.e., by replacing all \(x\)’s with \(x_{eq}\)’s (again, note that these could be vectors). This actually makes the left hand side zero, because \(x_{eq}\) is no longer a dynamical va... |
One can solve Poisson’s problem $-\Delta u = f$ in $d$ dimensions with homogeneous Dirichlet boundary conditions using a mixed formulation as explained below:
Let $\sigma = \nabla u$, then for a sufficiently smooth function $\tau$, by Green’s theorem
\begin{align*} (\sigma, \tau) &= (\nabla u, \tau) \\ &= -(u, \textrm{... |
It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in exponential time (for example, less than $2^n$ steps with only polynomial space)? E.g. in what time and space we can find a to... |
If the spherical approximation is good enough, you should be able to convert the surface areas into radii. I mention that because in my work with light scattering, all the equations are usually written in terms of the radius of the scatterers. It also looks like all the Mie scattering tables are written in terms of the... |
Someone showed me a derivation for the area of a circle today. They took a circle of radius $r$ and inscribed a regular polygon in the circle. If you take an $n$-sided polygon, then its area is:
$$\frac{1}{2}r^2\left(\sin{\frac{2\pi}{n}}\right)n$$
If you let $n$ go to infinity, then you get $\pi$$r^2$ as your area.
How... |
$S_n$ acts on $\mathbb{C}^n$ by permutation, and there are two conventions which work (permute basis vectors or permute components), but permuting basis vectors ends up being a bit more natural. Here is a little background first. Given a group $G$ and two sets $A$ and $B$ which $G$ acts on, then the set of functions $A... |
I think this is more probability theory question. The trick is, if $x \sim F(x) $ ($F(x)$ is the cumulative distribution function or CDF in abbreviation form) then the transformation $ u = F(x)$ produces the uniform random variable $u$ with uniform distribution on interval $[0,1]$. you can see link below for proof:
Sho... |
The lecturer taught this method in my Optimization and Control Theory Class and I wasn't quite there when he named it. Could you help me out?
He gave the following example of the method in class:
Example: Solve $$ \text{max} [ f(x) = x_1 (30 - x_1) + x_2 (50 -2x_2) - 3x_1 - 5x_2 - 10x_3]$$Subject to the constraints: $$... |
Mathematische Software – Geogebra
This is the Android Version of GeoGebra. It is a lot more cumbersome to use than the desktop version, but you can get used to it. Although I will be criticizing it a bit below it is a very nice software. It can do far more than an average student would ever need. If you do not know the... |
Ms Alexandra Kiňová (23) is expecting the first Czechia's naturally born quintuplets (a package of 5 babies) on Sunday morning (tomorrow; update: they're out fine) which would mean that we match the achievement of the most fertile U.S. state – Utah – from the last week. Cool anniversary:In late January, we celebrated t... |
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision difference_between_beta_coefficients_and_partial_correlation_coefficients [2019/07/12 23:21]
hkimscil
difference_between_beta_coefficients_and_partial_correlation_coefficients [2019/07/17 10:34] (current)
hki... |
When working with LaTeX, it is recommended to start each sentence on a new line. The reasons can be found in Axel Brandenburg’s computing tips and this stack overflow page so I won’t repeat them here. However, as an emacs user, I always want emacs do the formatting for me.
The existing solutions are summarized here. Th... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
There is a concept of
Rates of Growth in Calculus.
Definition :- f(x) <<< g(x) as x $\rightarrow$ $\infty$ means growth rate of f(x) is very slow as compared to g(x) when x $\rightarrow$ $\infty$ ie. $\frac{f(x)}{g(x)} \rightarrow 0 \, \, as \, \, x\rightarrow \infty$
It is true when f and g are non-negative.
So, we ca... |
I was trying to derive the Bernoulli equation from the above equation
for
time independent flow
If you are studying something
time independent then you just let $\frac{\partial}{\partial t}$ to be zero:
$$\frac{\partial}{\partial x_j} \left [ \frac{1}{2} \rho v^2 v_j + \rho h v_j + \rho \phi v_j \right ]=0$$
Next step ... |
In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case the worldsheet). The problem is that I don't understand exactly how the procedure works. I need to look at some worked e... |
Probabilistic bug hunting
Have you ever run into a bug that, no matter how careful you are trying toreproduce it, it only happens sometimes? And then, you think you've got it, andfinally solved it - and tested a couple of times without any manifestation. Howdo you know that you have tested enough? Are you sure you were... |
Regression (OLS) - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Regression (OLS)
$z$ test for the diff... |
Are there any analytical proofs for the 2nd law of thermodynamics?
Or is it based entirely on empirical evidence?
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.Sign up to join this community
It's simple to "roughly prove... |
If you assume that the $\lambda$-calculus is a good model of functional programming languages, then one may think: the $\lambda$-calculus has aseemingly simple notion oftime-complexity: just countthe number of $\beta$-reductionsteps $(\lambda x.M)N \rightarrow M[N/x]$.
But is this a good complexity measure?
To answer t... |
Image Dimensions Contents Describing the fields of the Canvas Properties Dialog
The user access the image dimensions in the Canvas Properties Dialog.
The 'Others' tab
Here some properties can simply be locked (such that they can't be changed) and linked (so that changes in one entry simultaneously change other entries ... |
I am solving the following exam problem.
Problem: An iterative scheme is given by $$ x_{n+1}= \frac{1}{5}\left(16-\frac{12}{x_n} \right).$$ Such a scheme with suitable initial approximation $x_0$ will
(a) not converge (b) converge to $1.6$ (c) converge to $1.8$ (d) converge to $2$
My attempt. By defining $g(x) = 1/5(16... |
Considering their distance from their parent stars, might Oort cloud object such as comets be exchanged between passing stars (assuming that other stars have similar Oort clouds)?
You can exclude the 'considering the distance' piece - of course Oort Cloud objects could transfer between different gravitational fields.
H... |
Forgive me in advance, this may get overcomplicated. I am going to give you the facts as scaled down as I can but still sufficiently detailed. I think providing you with what we have and allowing you to infer from it is the best way to avoid misrepresenting the answer.
Here is the General Relativity equation that descr... |
By the mean value theorem it's easy to show that $|a_{n+1}-a_{n}| \leq \frac{5}{6}|a_{n}-a_{n-1}|$ for every n.
Next, I thought of saying $|a_{n+1}-a_{n}| \leq ... \leq (\frac{5}{6})^{n}|a_{1}| \to 0$ and somehow show that ** if $M_{n}$ is the closed interval whose end points are $a_{n}$ and $a_{n-1}$ then $a_{n+1} \in... |
Suppose I have an Lagrangian $$\mathcal{L} = \frac{1}{2}g_{ab} \bar{\psi}^a \Gamma^k \partial_k \psi^b $$ and I want to show it's invariance under the infinitesimal Lorentz transformations $$\delta \psi^a = -\Lambda_{mn}x^m \partial^n \psi^a + \frac{1}{2} \Lambda_{mn} \Sigma^{mn}\psi^a ,$$ $$\delta \bar{\psi}^a = -\Lam... |
In Carroll's Appendix B, he says
You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with metric $g_{\mu \nu}$ and matter fields $\psi$, and $\phi : M \to M$ is a diffeomorphism, then the sets $(M, g_{\mu \nu}, \psi)... |
This section shows how to calculate the masses and moments of two- and three- dimensional objects in Cartesian \((x,y,z)\) coordinates.
Mass
We saw before that the double integral over a region of the constant function 1 measures the area of the region. If the region has uniform density 1, then the mass is the density ... |
In this section we are going to cover the integration of a line over a 3-D scalar field. When you learned on dimensional integrals, we integrated functions of \(y\) with respect to \(x\) and assumed that \(z\), the third dimension, does not change. If, however, the third dimension does change, the line is not linear an... |
SolidsWW Flash Applet Sample Problem 2
Line 327: Line 327:
== License ==
== License ==
−
The Flash applets are protected under the following license:
+
The Flash applets are protected under the following license:
[http://creativecommons.org/licenses/by-nc/3.0/ Creative Commons Attribution-NonCommercial 3.0 Unported Lic... |
I am building a model which predicts angles as output. What are the different kinds of outputs that can be used to predict angles?
For example,
output the angle in radians cyclic nature of the angles is not captured output might be outside $\left[-\pi, \pi \right)$ output the sine and the cosine of the angle outputs mi... |
"La Madre Terra" by Pietro Cascella, made for ICRANet. You can see Einstein's equation, which he translated into the metaphor "marble = wood". My guess is that it symbolizes Einstein's idea that everything (matter, life, not just the earth) emerges from the perfect geometry of spacetime. I took this photo at the Marco ... |
There are two kinds of vector multiplications-
Dot (Scalar) Product:
The dot product of two vectors gives a scalar, that means only the magnitude is left, no direction. Mathematically, it is equal to the product of the magnitude of two vectors times the cosine of the angle between the two. ie. $$\vec{v} \cdot \vec{u}=|... |
In some book about continuum mechanics I read that from principle of virtual work follows balance of rotational momentum when $\delta \boldsymbol{r} = \boldsymbol{\delta \varphi} \times \boldsymbol{r}, \; \boldsymbol{\delta \varphi} = \boldsymbol{\mathsf{const}}$ ($\boldsymbol{r}$ is location vector, $\delta \boldsymbo... |
Given $A_\infty$-spaces $X$ and $Y$, Boardman and Vogt defined an $A_\infty$-map from $X$ to $Y$ to be a map $f: X \to Y$ of underlying based spaces and an $A_\infty$-structure on the reduced mapping cylinder $$ M_f = Y \cup_{f\times 0} X\wedge (I_+) $$ which extends the given structures on $X$ and $Y$. These do not fo... |
62 0
Hi,
I'd like to know which property proves the following simple result.
Let p be a prime greatest than 3. r is a quadratic residue of p if there exists a such that: [tex]a^2 \equiv r \pmod{p}[/tex]. Since p is prime, there are [tex]\frac{p-1}{2}[/tex] different residues (not counting 0). Now, if you sum them all, ... |
Program Arcade GamesWith Python And Pygame
Searching is an important and very common operation that computers do all the time. Searches are used every time someone does a ctrl-f for “find”, when a user uses “type-to” to quickly select an item, or when a web server pulls information about a customer to present a customi... |
The problem is: prove the existence of function $f: \mathbb{R}\times\mathbb{R} \rightarrow \mathbb{N}$ such that $f(x,y)=f(y,z)\implies x=y=z$. I was thinking about $f(x,y) = \left\{ \begin{array}{ll} 1 & \textrm{iff $x=y$}\\ something & \textrm{ iff $x\neq y$} \end{array} \right.$ but I guess we need to have bijection... |
Suppose a quantum system (non-interacting) at finite temperature ($\beta^{-1}$). I want to know how to compute the transition probability between two degrees of freedom ($u$ and $v$) at two different times.
The system starts ($t=0$) in a mixed state, described by $$ \hat \rho = \sum_l e^{-\lambda_l \beta} |\psi_l\rangl... |
So the equation for the pressure within a non-rotating, spherical gas cloud of radius R and uniform density $\rho$ ( if it is in hydrostatic equilibrium) is :
$$P(r)=P_c-\tfrac{2\pi}{3}G\rho^2r^2$$
starting from the hydrostatic equation in spherical co-ordinates
$$\tfrac{dP}{dr}=-\tfrac{GM(r)}{r^2}\rho(r)$$ we find tha... |
\[ y=f(x)=e^{-5x^2} , 0\leq x \leq 1\]
Figure 15.1-0
\[ \int_a^b f(x)\;dx = \lim_{n\rightarrow\infty}\sum_{i=1}^n f(x_i)\, \Delta x_i \]
A fundamental method to calculate the area: the base of function f(x) is equally divide into n pieces whose width are \( \Delta x \). Then \( S=\Delta x f(x_i) \) is the area of the r... |
Recall that the chain rule states that
\[ (f(g(x)))' = f'(g(x))g'(x). \]
Integrating both sides we get:
\[ \int[f(g(x)]'dx = \int[f'(g(x)g'(x)dx]\]
or
\[ \int f'\left( g(x) \right) \, g' (x) \, dx = f\left(g(x)\right) + C \]
Example 1
Calculate
\[ \int \dfrac{2x}{x^2+1}\, dx = \int 2x\left( x^2+1\right)^{-2} \, dx. \]
... |
Focus is $(2,3)$, point on directrix is $(-3,2)$. Parabola touches $x$-axis. Find vertex.
I would be thankful if someone could help me with this problem.
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.... |
I am working on a linear analysis problem where we have boiled down the problem to finding a continuous function $f:\mathbb{R} \to \mathbb{R}$ that is bounded, but has infinite derivative at zero. So far, we have conjured up the example $$f_n(x) = \frac{2}{\pi}\arctan(nx)$$ This sequence of functions will have infinite... |
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the absolute Galois group of $\mathbf{Q}$. Moreover, fix a rational base point $x$ in $X(\mathbf{Q})$.
Let $\sigma:G_{\mathbf... |
Suppose $f: A \to B$ and $g: B \to A$ are injections of rings(
commutative with identity). Must $A$ and $B$ be isomorphic asrings?
According to this question, this answer should be "no", but can someone give an example?
Thanks!
MathOverflow is a question and answer site for professional mathematicians. It only takes a ... |
Say we want to compute the Coleman-Weinberg potential at 2 loops.
The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + \phi$, and do a path integral over the quantum part of the field, $\phi$.
We can retrieve the effective action by doing a path... |
In
sequent calculus LK (see Gaisi Takeuti, Proof Theory (2nd ed - 1987)) we have a "standard" derivation of Double Negation in the form $\rightarrow \lnot \lnot A \supset A$.
We have to start from an
Axiom :
$$\frac{A \rightarrow A}{\rightarrow \lnot A, A}$$ by $\lnot$-right;
then :
$$\frac{\rightarrow \lnot A, A}{\lno... |
(From the comment above)The problem seems coNP-hard; the simple reduction is from 3CNF-UNSAT (which is coNP-complete):given a 3CNF formula $\varphi = C_1 \land ... \land C_m$, extend it adding a new clause with 4 new variables:
$$\varphi' = (y_1 \lor y_2 \lor y_3 \lor y_4) \land C_1 \land ... \land C_m$$
$\varphi'$ has... |
How can I calculate the following limit:
$$ \lim_{n\to \infty} \frac{2\cdot 4 \cdots (2n)}{1\cdot 3 \cdot 5 \cdots (2n-1)} $$ without using the root test or the ratio test for convergence?
I have tried finding an upper and lower bounds on this expression, but it gives me nothing since I can't find bounds that will be "... |
LaTeX:Symbols
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - Commands - Packages - Help
This article will provide a short list of commonly used LaTeX symbols.
Contents Finding Other Symbols
Here are some external resources for finding less commonly used s... |
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... |
Answer:
Given, radius of base \[r=3.5\text{ }cm\] Total height of toy \[=15.5\text{ }cm\] Height of cone \['h'=15.5-3.5\] \[=12cm\] Slant height \['l'=\sqrt{{{h}^{2}}+{{r}^{2}}}\] \[=\sqrt{{{12}^{2}}+{{3.5}^{2}}}\] \[=\sqrt{144+12.25}\] \[=\sqrt{156.25}\] \[=12.5\,\,cm\] Total S.A. of toy = CSA of cone + CSA of hemisph... |
We now come to the first of three important theorems that extend the Fundamental Theorem of Calculus to higher dimensions. (The Fundamental Theorem of Line Integrals has already done this in one way, but in that case we were still dealing with an essentially one-dimensional integral.) They all share with the Fundamenta... |
Given arbitrary sets $A$ and $B$, the notation $A^{B}$ is mostly clear from context to mean $A^{B} = \{f : f : B \rightarrow A\}$.
However, when these sets are ordinal or cardinals, especially $\omega$, the notation is not consistent even among subfields of logic.
For example $2^\omega$ in one sense can denote ordinal ... |
Can we upgrade the Reflection axiom schema in Ackermann to the following:
Modified Reflection axiom schema: if $\psi(y)$ is a formula that doesn't use the symbol $V$, in which only symbols $y,x_1,..,x_n$ occur free, and in which $x$ is not free, then all closures of:
$$x_1,..,x_n \in V \wedge \forall y (\psi(y) \to y \... |
I have a (I hope) simple question! If I had a linear regression,
$Y_t = \alpha + \beta X_t + \epsilon_t$
with $\epsilon_t \sim N(0,\sigma^2)$
and I assume a Cauchy prior for $\sigma$, is it possible to get a conditional conjugate posterior, that I could embed in a Gibbs sampler without having to rely on a MH step? I am... |
Program Arcade GamesWith Python And Pygame
Searching is an important and very common operation that computers do all the time. Searches are used every time someone does a ctrl-f for “find”, when a user uses “type-to” to quickly select an item, or when a web server pulls information about a customer to present a customi... |
num2vec: Numerical Embeddings from Deep RNNs Introduction
Encoding numerical inputs for neural networks is difficult because the representation space is very large and there is no easy way to embed numbers into a smaller space without losing information. Some of the ways to currently handle this is:
Scale inputs from m... |
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