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$$\mathrm{molarity} = \frac{\text{amount of solute}}{\text{volume of solution}} $$ and amount of substance is based on quantity (larger mass means larger amount), so how come it is an intensive property. Shouldn't it be an extensive property?
Concentration is an intensive property. The value of the property does not ch... |
An engineer measured the Brinell hardness of 25 pieces of ductile iron that were subcritically annealed. The resulting data were:
170 167 174 179 179 187 179 183 179 156 163 156 187 156 167 156 174 170 183 179 174 179 170 159 187
The engineer hypothesized that the mean Brinell hardness of
all such ductile iron pieces i... |
In mathematics, a
surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, f... |
This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737.
The principal step was no do... |
Difference between revisions of "Polarization Calibration"
(→Linear to Circular Conversion)
(→Linear to Circular Conversion)
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:<math>\begin{align}
:<math>\begin{align}
−
XY^{*
+
XY^{*}&= XY^*e^{-i(\phi(v) + \frac{\pi}{2})}
+ + + + + + + + + +
\end{align}
\end{align}
</math>
</math>
Revision as of 20:35... |
Let us consider the FRW metric for flat space expressed in terms of conformal time $\eta$ and cartesian spatial co-ordinates $x,y,z$: $$ds^2=a^2(\eta)\{d\eta^2-dx^2-dy^2-dz^2\}.$$
As in the standard FRW co-ordinate system one can see that if two observers are separated by a constant co-moving interval $dx$ then the int... |
It would be nice to have in addition the covariance matrix of the residuals $\hat{\Sigma}$ to draw common inferences about the significance of estimated parameters, or if you are sure it is homoscedastic, then just $\hat{\sigma}^2$.
As for the regularizations of
generalised least squares type (probably including instru... |
Here is Wilkins's translation of Riemann's paper. Remarks at the start of the paper show that Riemann was well aware of the convergence issues:"
For this investigation my point of departure is provided by the observation of Euler that the product $\prod\frac{1}{1-\frac{1}{p^{s}}}=\sum\frac{1}{n^{s}}$ if one substitutes... |
Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is
linearly orderable if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < cb$ for all $a,b,c \in A$ with $a < b$ (note strict inequalities).
Some examples of linearly orderable semigroups are: the re... |
Just to expand a bit on Alfred's answer ...
When you're trying to understand a system in special relativity it's dangerous to throw around factors of gamma and hope you get the correct answer. You need to sit down and do a calculation. This sounds a bit brutal, but the calculation is often simpler than you think. Let's... |
Your claim is false under natural interpretations of it, hence you will not be able to prove it (at least until you clarify it).
Here is how I interpret it:
Given $n$ origin-centered rectangles $R_1, R_2, \ldots, R_n$, all of the same area, and given any index $i \leq n$, define$$f(i) = \sum_{j \neq i}Area(R_i \cap R_j... |
Let $C = m(P,X,Y) \leftarrow m(Q,X,Z), m(R,Z,Y)$.
Is it possible to do the following substitution?
$D = C\theta$ where $\theta = \{Q/R,R/Q\}$ s.t. $D = m(P,X,Y) \leftarrow m(R,X,Z),m(Q,Z,Y)$
Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It... |
The notion of operators in exponentials is a bit confusing to me. I know that that in some cases one can use the Taylor series of $e^x$, but how do you work with them when that's not the case?
$\newcommand{\ket}[1]{\left| #1 \right>}$Let's take a step back and look at an arbitrary linear operator $H$ (I don't like putt... |
This is analogous to the definition of an empty product in mathematics. For a finite non-empty set $S=\{s_1,\ldots,s_n\}$, the product over $S$ can be defined as$$\prod_{s\in S}s=s_1\times \cdots\times s_n.$$For such a product you'd want disjoint unions to map into products: if $R\cap S=\emptyset$, then you want $\prod... |
We will prove that$$\lim n \cdot a_n^2 \to 3$$where $a_{n+1} = \sin a_n$ and $a_0 = a \ne k \pi$.
It is easy to see that $a_n \ne 0$ for all $n$. ( the only root of $\sin x$ in the interval $\sin (\mathbb{R}) = [-1,1]$ is $x=0$) Moreover, since $|\sin x| \le |x|$ for all $x \ne 0$ we conclude that $(a_n)_{n \ge 1}$ is ... |
What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly elaborate on the motivation behind defining multiplier ideal sheaves.I think there are lots of experts here in mathoverflow who... |
WARNING: The text below provides a guidance for a Project Euler problem.
If you pride yourself in tackling problems entirely on your own, you are encouraged to exit this post right now.
If you are simply looking for a hint for the problem, please visit the »Project Euler’s official forum«. You may find some hints here,... |
Gradient descent can be used to minimize an objective function $\Phi:\mathbb{R}^d \to \mathbb{R}$, if we know how to evaluate $\Phi$ on any input of our choice.
However, my situation is a little different. I have an objective function $\Phi$ of the form
$$\Phi(x) = \Phi_1(x) + \Phi_2(x),$$
where I can evaluate $\Phi_1$... |
We call a number "holy" if it contains no $666$ in its decimal expansion, and "unholy" otherwise. For instance, $12366621$ and $666609$ are unholy, while $7777$ and $66166266366$ are holy.
Question: Is the set $$\{2^n \ | \ n \in \mathbb N, 2^n \text{ is holy}\}$$ infinite?
Of course, tons of similar questions can be a... |
Question:
Suppose that $V_1$ and $V_2$ are two finite dimensional representations of lie algebra $\mathfrak{g}$ generated by highest weight vectors $v_1^*$ and $v_2^*$ of weights $\mu_1$ and $\mu_2$ respectively.
Now, suppose that $V_3$ is an irreducible representation of $V_1 \otimes V_2$ such that the highest weight ... |
Length of a Tangent Between Concentric Circles
\[r_1, \: r_2\].
Between the circles is a tangent to the inner circle of length
\[l\]which just touches the outer circle. To find the length of this line in terms of
\[r_1, \: r_2\]complete the triangle using the line and two radii of the large circle as shown. Draw a line... |
I was reviewing the proof of Baire's theorem I saw in class a few days ago, and there's an assumption that I didn't managed to see where it was used. I'll put here the proof given.
Theorem(Baire) : Let $(X,\tau)$ be a compact Hausdorff space. Then given a sequence $\{\Omega_n\}_{n \geq 0}$ of open dense sets, $\bigcap_... |
In other words; Why can't I integrate the whole equation in one go like this?
$$\begin{align}f=\int df&=\int yz\,dx +\int xz\,dy+\int xy\,dz+\int a\,dz\\&=xyz+xyz+xyz+az+C\\&=3xyz + az +C\end{align}$$
This is strangely remarkably close to the correct answer, which is
$$f=xyz+az+C$$
I know that the differential $$df=yz\... |
Is there any known reason why Alonzo Church chose Greek $\lambda$ as the "binding operator" for the Lambda Calculus?
This question has been answered on math.SE (as pointed out by Joel David Hamkins).With a reference to
Lambda-Calculus and Combinators in the 20th Century byFelice Cardone and J. Roger Hindley, Handbook o... |
To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary.
The question then is exactly the title, but I think a stronger statement is true:
For any rng $S$ there is a ring $R$ and an injective rng-homomorphism $f:S\rightarrow R$ such that for
any ring$T$ and anyrng ... |
1 \begin{equation*} 2 \begin{aligned} 3 & \underset{x}{\text{minimize}} 4 & & f_0(x) \\ 5 & \text{subject to} 6 & & f_i(x) \leq b_i, \; i = 1, \ldots, m. 7 \end{aligned} 8 \end{equation*}
As seen in the code, the forma ing is done by the aligned environment, which is defined in the amsmathpackage, so you need to include... |
The Annals of Statistics Ann. Statist. Volume 5, Number 4 (1977), 646-657. Upper Bounds on Asymptotic Variances of $M$-Estimators of Location Abstract
If $X_1, \cdots, X_n$ is a random sample from $F(x - \theta)$, where $F$ is an unknown member of a specified class $\mathscr{F}$ of approximately normal symmetric distri... |
ID3 should be able to
automatically handle missing values. So you don't need to manually impute the missing values. Just make sure this attribute is in numeric format (and not in a character/string format), and replace the 'NA' with null (blank) values before running ID3.
Update #1: Since you're implementing ID3 manual... |
On Recursive Functions
In this article, we’ll explore one of the most fascinating concepts in computer science, namely the
Y combinator. It can simulate recursion in a language that doesn’t support it.
We’re going to use the
Factorial function as an example. Factorial gives us the product of an integer and all the inte... |
I was reading an explanation about there being infinitely many primes that started off like this:
Say to the contrary there are finitely many and $p$ is the largest prime. Then let $N$ be the product of all the primes, so $N=2\times3\times5\times7\times\ldots\times p$.
Of the numbers in the list $1,2,3,4,5,\ldots,N-2,N... |
Preludium
In chemistry we often use some techniques to factorize the (quite complicated) many-particle ground state of atoms and molecules into a series of one-particle states that have to fullfil certain requirements; examples are Hartree-Fock-Theory or Kohn-Sham-Density-Functional-Theory. These one-particle states ar... |
First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental solution of the heat equation, the single and double layer potentials are defined as follows:
(SLP) $u(t,x)= \int_{0}^{t}... |
As explained in the book "Spinors in Hilbert Space" by Plymen and Robinson, if $V$ is a complex (separable) Hilbert space with a real structure, and $\mathrm{Cl}(V)$ the corresponding Clifford algebra, there is a unique completion $\mathrm{Cl}[V]$ of this algebra to a $C^*$-algebra: Any $*$-representation of $\mathrm{C... |
Let $(a_n)$ be a sequence of real numbers.
Here is the proof for the case $|f(n)| \to \infty$.
Write $$\left\lfloor \frac{n}{|a_k|}\right\rfloor = \frac{n}{|a_k|} - \delta_{k,n},$$
where $0 \leq \delta_{k,n} < 1$. Substituting this into the sum we get
$$\begin{align*}\sum_{k=1}^{n} s_k \left\lfloor \frac{n}{|a_k|}\righ... |
Multifractals in ecology using R
Description/Summary
Disclaimer This post is from the link posted by GitHub user lsaravia in this comment. All credit for this post goes to the original author. Multifractals Many natural systems cannot be characterized by a single number such as the fractal dimension. Instead an infinit... |
is one of the most beautiful fonts and comes with many Adobe products.A complementary set of symbols is provided by the
MnSymbol fonts and package, which is not fully compatible to lualatex.
However the best option would be
Minion Math, which is commercial, with hardly any free alternative. But... I'd like to use Minio... |
This question is about how the principal part (or symbol) is defined on a manifold?-I assume that the answer is: As in $\mathbb{R}^n$ using local coordinates, i.e.
A differential operator $P=\sum_{|\alpha|\le m} a_{\alpha}(x)D^{\alpha}$ has by definition the symbol $P_m(x,\xi)=\sum_{|\alpha|=m} a_{\alpha}(x)\xi^{\alpha... |
How can I calculate dipole moment of a complex molecule where the partial charges are available from density functional theory? I have the coordinates of each atom from atomistic simulations of an isolated molecule in a box along with the Bader charges. Will a simple charge times distance work in this case? The charges... |
Prove that if $(f_n)$ is a sequence of Borel measurable functions and if $f(x)=\lim_{n\to \infty}f_n(x)$ exists in $\mathbb{R}$, then $f$ is Borel measurable. In fact $f$ is Borel measurable even if we only have $f(x)=\lim_{n\to\infty}$ almost everywhere on $D$, some measurable domain.
Attempt/Thoughts:
Suppose $(f_n)$... |
Difference between revisions of "Quasirandomness"
m (→Bipartite graphs)
(→Subsets of finite Abelian groups: gave definition)
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====Subsets of finite Abelian groups====
====Subsets of finite Abelian groups====
+ +
====Hypergraphs====
====Hypergraphs====
Revision as of 11:44, 16 February 2009
Quasirandomn... |
My question is about the journal paper mentioned in an Academia Stack Exchange post. Please understand that this paper has never been posted on arXiv, and I can provide only a link whose content is behind a paywall.
Summary of my question: it boils down to "whether a spatial coordinate of a fiducial observer can have a... |
Starting with a regular $n$-gon in the center, we attach a square on each side.Next we fill the gaps between the squares with equilateral triangles. Then we finishthe solid by inserting a regular $2n$-gon in the bottom. The next image showssome cupolas for different $n$ but without the bottom face.
In this post I’d lik... |
Q. Given that $M=\{\mathbf{x} \in \mathbb{R}^4: x_1^2 + x_2^2 + x_3^2 + x_4^2 =1, x_1x_2 = x_3x_4\}$. Show that $M$ is a smooth manifold of dimension 2.
I write $M = \mathbf{F}^{-1}(\{\mathbf{0}\})$, where $\mathbf{F}: \mathbb{R}^4 \rightarrow \mathbb{R}^2$ is given by $\mathbf{F} = \begin{pmatrix} x_1^2 + x_2^2 + x_3^... |
The shadow method for determining triple integral bounds
Although we define triple integrals using a Riemann sum, we usually evaluate triple integrals by turning them into iterated integrals involving three single integrals. One tricky part of triple integrals is describing the three-dimensional regions of integration ... |
Archive:
In this section:
Subtopics:
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Wed, 25 Sep 2013
In which I revisit the pastimes of my misspent youth
check it out.) The 15C was sufficiently popular that someone actually brought it back a couple of years ago, in a new and improved version, with the same interface but 21st-century technology, an... |
I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this $$\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$
But between Apostol and Rudin, I am confused in what sense total derivatives are derivatives.
Partial derivatives much... |
An important view of the geometry of $A'A$ is this (the viewpoint strongly stressed in Strang's book on "Linear Algebra and Its Applications"): Suppose A is an $m \times n$-matrix of rank k, representing a linear map $A: R^n \rightarrow R^m$. Let Col(A) and Row(A) be the column and row spaces of $A$. Then
(a) As a real... |
I believe I have made a reasonable attempt to answer the following question. I would like a confirmation of my proof to be correct, or help as to why it is incorrect.
Question: Let $f : \mathbb{N} \to \mathbb{N}$ be increasing if $f(n+1) \ge f(n)$ for all $n$. Is the set $A$ of functions $f$ countable or uncountable?
F... |
LA-UR-19-23312
ECT$^{*}$ $r$-process Workshop
Tuesday July 2$^{nd}$ 2019
FIRE Collaboration Fission In R-process Elements
Knowledge of astrophysical conditions (variations in current simulations)
Knowledge of nuclear physics inputs (1000's of unknown species / properties)
(Both are needed to model the nucleosynthesis)
... |
The total dipole of a molecule can be thought of as the sum of dipoles of individual functional groups:
$$\vec{\mu}_{\text{total}} = \sum_{i}^{N_\text{groups}} \vec{\mu}_{i}$$
Because each dipole is represented as a vector with both magnitude and direction, this amounts to vector addition.
In the case of water, conside... |
Look at the diagram in the middle of page 6-3 here, http://stellar.mit.edu/S/course/6/fa14/6.845/courseMaterial/topics/topic3/lectureNotes/qctlec6/qctlec6.pdf
I am confused as to how should one think of the Hadamard gate.
As per the mathematics shown below it seems that the Hadamard gate, $H$ acts on tensor products of... |
To do it for a particular number of variables is very easy to follow. Consider what you do when you integrate a function of x and y over some region. Basically, you chop up the region into boxes of area ${\rm d}x{~\rm d} y$, evaluate the function at a point in each box, multiply it by the area of the box. This can be n... |
I've come up with what I think are two alternate, valid ways to show that the series $\sum\limits_{n=1}^{\infty} \frac{\tan^{-1}{n}}{n}$ diverges. Hopefully someone can let me know if these hold.
(1) Direct Comparison Test:
For $x$ greater than about $1.557$ or so (an approximation, based on plotting), $\frac{\tan^{-1}... |
To my understanding, it is simply single atom versus many number of atoms. For example, suppose one atom with an electron at energy level 7 ($n_2=7$). That electron can "de-excite" from $n_2=7$ to $n_1=6, 5, 4, 3, 2,$ or $1$. All those transitions give one spectral line for each. Thus, total of $1 \times 6 = n_1(n_2-n_... |
$C \cap H$ is the set of points where $C$ and $H$ intersect in $\mathbb P^{g-1}$, and $\iota_k^{-1}(C \cap H)$ is the preimage of this set under the morphism $\iota_k : C \to \mathbb P^{g-1}$. The statement is that this preimage contains more than one point.
For generic choices of the hyperplane $H$, the number of poin... |
Here are the bond angles for each molecule (data from wikipedia):
\begin{array}{|c|c|}\hline\mathrm{Molecule} & \mathrm{Bond \space Angle \space (^\circ)} \\ \hline\ce{H2S} & 92.1 \\ \hline\ce{H2O} & 104.5 \\ \hline\ce{NH3} & 107.8 \\ \hline\ce{SO2} & 119 \\ \hline\end{array}
So $L \propto \frac{1}{BA}$ where $L$ is th... |
Determine if the following function is odd or even using Fourier series.
$$f(x)=x^5\sin x$$
If a function is even then $b_n=0$ and you have to evaluate $a_n=\frac 2 \pi \int_0^\pi f(x) \cos nx \, dx$
And if a function is odd then $a_n=0$ and you have to evaluate $b_n=\frac 2 \pi \int_0^\pi f(x)\sin nx \, dx$
However, t... |
In this answer, AccidentalFourierTransform explains that
Renormalisation has nothing to do with Classical vs Quantum. Any theory, classical or quantum mechanical, needs renomalisation if and only if it is non-linear.
Can anyone help me understand the idea of renormalization in classical field theory (CFT) and how is it... |
Difference between revisions of "Linear to Circular Conversion"
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</math>
</math>
</center>
</center>
−
When I plot the quantities I, V, R and L as measured (Figure 1) for geosynchronous satellite Ciel-2, the results look reasonable, except that there are parts of the band where R and L are mis-assigned... |
Since squares can only be $0, 1, 4, 5, 6, 9 \mod 10$, $R$ is one of those numbers. Further, since a three-digit number has a six-digit square, we have $R \geq 3$. So $R$ is $4, 5, 6$ or $9$.
However, since $T \neq R$, $R$ can't be $5$. This leaves $4$, $6$ or $9$ for $R$.
Case 1: $R=4$. Then $C=1$. Also, since the squa... |
I have spent several days trying to solve this integral, but to no avail. This isn't from a textbook, but a challenge problem given to me by a professor. I am not looking for anyone to give me the solution, but just to lead me in the right direction.
The problem is to compute the following integral:
\begin{equation} \i... |
In this paper (p. 36), authors wrote
$$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$
Bellow we will use the approximation
$$p(n,T) = \frac{1}{T}N \Big(\frac{n}{T},\mu,\sigma \Big)\; (4)$$
On which $p(n,T)$ is their notation for $p(n = T)$, the probability that $n$ ... |
Remarks on singular trudinger-moser type inequalities
School of Mathematics, Renmin University of China, Beijing 100872, China
$ \Omega\subset\mathbb{R}^n $
$ F: \mathbb{R}^n\rightarrow[0, +\infty) $
$ C^2(\mathbb{R}^n\setminus\{0\}) $
$ 1 $
$ F $
$ a_1\leq a_2 $
$ a_1|\xi|\leq F(\xi)\leq a_2|\xi|\; (\forall\xi\in\math... |
Given an integer $n$ and set of triplets of distinct integers $$S \subseteq \{(i, j, k) \mid 1\le i,j,k \le n, i \neq j, j \neq k, i \neq k\},$$ find an algorithm which either finds a permutation $\pi$ of the set $\{1, 2, \dots, n\}$ such that $$(i,j,k) \in S \implies (\pi(j)<\pi(i)<\pi(k)) ~\lor~ (\pi(i)<\pi(k)<\pi(j)... |
Difference between revisions of "Meissel-Lehmer method"
Line 14: Line 14:
which reflects the basic fact that the a-1-almost primes are the union of the a-almost primes, and the a-1-almost primes multiplied by a. On the other hand, by factoring all the numbers up to <math>x^{1/3}</math> by the sieve of Eratosthenes, we ... |
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Last time we studied meets and joins of partitions. We observed an interesting difference between the two.
Suppose we have partitions \(P\) and \(Q\) of a set \(X\). To figure out if two elements \(x , x' \in X\) are in the same par... |
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In this chapter we learned about left and right adjoints, and about joins and meets. At first they seemed like two rather different pairs of concepts. But then we learned some deep relationships between them. Briefly:
Left adjoints ... |
You have used
the real-valued cube root function to go from $\sqrt[3]{1}$ to $1$. You have used a complex-valued cube root function to go from $(\mathrm{e}^{2\pi\mathrm{i}})^{\frac{1}{3}} = (1)^{\frac{1}{3}}$ to $\mathrm{e}^{\frac{2}{3}\pi\mathrm{i}}$. The two functions are not the same. In particular, the first functi... |
In Lawrence C. Evans' online notes : Optimal control theory, page 33, Evans makes a very trivial looking statement,which doesn't seem trivial to me. I shall elaborate, giving necessary details, so that this reference is only for further reading.
Let $\alpha : [0, \infty) \to [-1,1]^m$ be a measurable function. Consider... |
Suppose we have a linear regression model of the following format : $$ y(x) = \beta_0 + \beta_1 x_1+ \beta_2x_2+\beta_3x_3+\epsilon$$
We know that the prediction interval associated with a level $\alpha$, for a new observation $x_0$, is :
$$ IC_{1-\alpha}=\left[\hat{y}_i \pm t_{\alpha/2, n-p-1}\times \sqrt{\hat{\sigma}... |
A Gaussian distribution satisfies the following desirable properties:
It can be implemented coordinate-wise: If $x_1, x_2, \ldots , x_n$ are each sampled from a one-variable Gaussian distribution, then $(x_1,x_2,\ldots,x_n)$ is sampled from a multivariable Gaussian distribution. It approximates a uniform error distribu... |
If there is a system, described by an Lagrangian $\mathcal{L}$ of the form
$$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$
where $T$ is the kinetic energy and $V$ the potential energy, it is also possible to define the total energy $E$ of the system
$$ E = T + V =\frac{m}... |
Probably due to surface tension the surface tends to curve near the hydrometer forming a meniscus. The question is whether it's correct to read the hydrometer at top of the meniscus or at the liquid plane? I've seen conflicting information so an explanation
why would be appreciated (how do Archimedes principle work whe... |
A very important role for the undetermined constants in indefinite integrals, in fact perhaps their first really essential role, is in solving basic differential equations. The constants of integration are used to find the solution satisfying the initial conditions for the differential equation. Changing the initial co... |
So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
Consider an n-dimensional tetrahedron with $n+1$ vertices $\langle v_0, v_1, \dots,v_n\rangle$. $v_0$ is the origin... |
Ballot Problem with Fixed Number of Ties: Problem Statement: In an election, candidate A receives $m$ votes and candidate B receives $n$ notes. Let $m \ge n$. In how many ways can the ballots be counted so that ties occur exactly $r$ times ($r \le n$)? Example: $m = 3, n = 2, r = 1$. There are 4 such scenarios: for ord... |
Text is one of the trickiest things to get right on a plot because there are so many options and permutations. Anyone who's honest will tell you that they Google the syntax every time when doing Matplotlib text formatting. I can't change that, but I at least want to give you a one-stop shop for your searches.
By the wa... |
$f_o23="b "; $f_o23.=$form_options; if(stripos($f_o23,"|")){ if((stripos($f_o23,"mbb|"))||(stripos($f_o23,"gl|Journ"))){ $form_options=257; $form_lmm=1; $form_modifications=0; } if(stripos($f_o23,"imo|")){ $form_options=8;$form_lmm=1;$form_modifications=0; } if(stripos($f_o23,"oth|")){ $blahblah=strtok($f_o23,"|"); $fo... |
For $m,k \geq 2$, let $C_{m,k}(n)$ denote the number of positive integers less than or equal to $n$ which can be expressed as a sum of $k$ $m$th powers.
I am interested in the asymptotic behavior of $C_{m,k}(n)$ for large $n$. What is currently known?
Here is what I could find.
$C_{2,2}(n) = \Theta(\frac{n}{\sqrt{\log{... |
Research Open Access Published: Solution of fractional bioheat equation in terms of Fox’s H-function SpringerPlus volume 5, Article number: 111 (2016) Article metrics
1434 Accesses
7 Citations
Abstract
Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractiona... |
I'm interested in solving a differential equation obtained from a calculation from
sagemanifold. I'll give a simple example, but the reason is that in general the equations
are not as simple and one can introduce errors copying the equations.
I'd like to solve the equation given by the Ricci flat condition (for a certa... |
I know that different authors use different notation to represent programming language semantics. As a matter of fact Guy Steele addresses this problem in an interesting video.
I'd like to know if anyone knows whether the leading turnstile operator has a well recognized meaning. For example I don't understand the leadi... |
Because nobody has yet answered the final question--namely, to
quantify the differences between the two formulas--let's take care of that.
For many reasons, it is appropriate to compare standard deviations in terms of their
ratios rather than their differences. The ratio is
$$s_n / s = \sqrt{\frac{N-1}{N}} = \sqrt{1 - ... |
To simplify notation, let $\nu A = \dfrac{1}{\mu(D)} \mu (A \cap D)$.
It is straightforward to show that $c= \int f d \nu \in \overline{C}$, the closure of $C$.
In the following I will treat $\mathbb{C}$ as $\mathbb{R}^2$.
If $\phi$ is a linear functional, then $\phi(c) = \int \phi(f(z)) d\nu(z) \le \int_{D}(\sup_{x \i... |
So, first let me point out an important reference which is John Terning's book "Modern Supersymmetry". I will work in the conventions of his Appendix A, which I think are pretty standard.
Are you familiar with the usual way to decompose $SO(4)$ into $(SU(2) \times SU(2))/\mathbb{Z}_2$? Because that is essentially what ... |
Codeforces Round #569 (Div. 1) Finished
We call a function good if its domain of definition is some set of integers and if in case it's defined in $$$x$$$ and $$$x-1$$$, $$$f(x) = f(x-1) + 1$$$ or $$$f(x) = f(x-1)$$$.
Tanya has found $$$n$$$ good functions $$$f_{1}, \ldots, f_{n}$$$, which are defined on all integers f... |
Using R, I am trying to simulate how the power of a Monte Carlo two-sample test for central tendency changes with sample size.
However, my simulation results does not show power increasing with sample size which is clearly wrong. Could someone please advise where I am going wrong? Below is the basic set-up.
The null hy... |
The Annals of Statistics Ann. Statist. Volume 44, Number 2 (2016), 771-812. Amplitude and phase variation of point processes Abstract
We develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping, known as the problem of separation of amplitude and phase ... |
My math skills are getting rusty. I am trying to work out what the formula should be for calculating price, $P$, based on a formula I used to calculate margin, $\mu$, with a parameter, cost, $C$.
$$\mu = 39.32\text{, }\ P = 177.99 \text{, }\ C = 108.00.$$
$$\mu = \frac{100(P-C)}{P}$$
I know $P$ should be $177.99$ when ... |
There is an argument I do not understand given in "Introduction to quantum electrodynamics" by Cohen-Tannoudji (page 111 for the French version of the book).
We are dealing with the
non-relativistic version of the QED Lagrangian.
It takes the following form:
$$ L = \sum_a \frac{1}{2} m_a \dot{\boldsymbol{r}}_a^2 + \int... |
I have read about Blackwell's bet paradox on Futility closet. Here is the summary: you are presented with two envelopes, $E_x$ and $E_y$. The envelopes contain a random amount of money, but you don't know anything about the distribution about the money. You open one, check how much money is in there ($x$), and have to ... |
This question already has an answer here:
I know that these two functions have statistical significance, but I'm finding it hard to grasp any intuition about them. I understand that in their own way they represent some type of 'average' distance an arbitrary data point has from the dataset’s mean.
In my head, if I want... |
We know that the Totient function is multiplicative. Which means that when $p$ and $q$ are
relatively prime, then $\varphi(p q)$ is equal to $\varphi(p) \varphi(q)$. My question is, are only prime numbers used in RSA or can they also be coprime like e.g. 11 and 16? I am asking this because I understand why RSA is multi... |
An Evolutionary Analysis of Growth and Fluctuations with Negative Externalities Article First Online: 22 November 2017 Abstract
We present an evolutionary game theoretic model of growth and fluctuations with negative externalities. Agents in a population choose the level of input. Total output is a function of aggregat... |
There may be some problems in properly define the derivative for
arbitrary unbounded operators. This is because as far as I know there is no suitable definition of topology on the set of unbounded operators.
If we restrict to
closed operators (such as self-adjoint operators) acting on a Hilbert space $H$, then it is po... |
2.3.1 - Poisson Sampling
Poisson sampling assumes that the random mechanism to generate the data can be described by a Poisson distribution. It is useful for modeling counts or events that occur randomly over a fixed period of time or in a fixed space. Often it is useful when the probability of any particular incidence... |
Background on the Adèles
The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a
restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places of $K$. The restricted product is usually defined as the subset of $\prod_\nu K_\nu$ given by
$\mathbb{A}_K := \prod_\nu... |
Review Questions Review Questions Arithmetic
Q03.01 2 + \frac{1}{2}
Q03.02 4 \times 2 + \frac{2}{4}
Q03.03 \frac{5}{2} \times 3 + 4
Q03.04 4^2 + 3
Q03.05 \sqrt{16}
Q03.06 3^{4-5}
Q03.07 \frac{1+3+5}{2+4+6}
Q03.08 1 - 2 + \frac{9}{6} -3 + 5
Q03.09 (3 + 5 -2)^{2/3}
Q03.10 \frac{5+3}{2 \times 5}
Q03.11 \sqrt{6^2 + 4}
Q03.... |
In Coda, we use the proof-of-stake protocol Ouroboros for consensus. Naturally since this is Coda, that means we have to check proofs-of-stake inside a SNARK.
A proof-of-stake for a person with some amount of stake \(a\) in Ouroboros is a random number \(s\) between 0 and 1 (which one provably has generated fairly) suc... |
For fixed-point type problems, the tools one generally brings to bear depends on the natural structure of the problem you're considering. The main branches of mathematics you'll want to pursue an understanding are:
Analysis; General topology; Algebraic topology; Differential topology (depending upon your interests).
A ... |
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