text stringlengths 256 16.4k |
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$\tan{(A+B)}$ $\,=\,$ $\dfrac{\tan{A}+\tan{B}}{1\,–\,\tan{A}\tan{B}}$
$\dfrac{\tan{A}+\tan{B}}{1\,–\,\tan{A}\tan{B}}$ $\,=\,$ $\tan{(A+B)}$
The tan of angle sum identity is called as tan of sum of two angles identity or tan of compound angle identity. It is mainly used in mathematics in two cases possibly.
The tan of a... |
Combination Theorem for Cauchy Sequences/Quotient Rule Theorem
Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $R$.
Suppose $\sequence {y_n}$ does not converge to $0$.
Then:
$\exists K \in \N : \forall n > K : y_n \ne 0$.
and the sequences
$\sequence { {x_{K+n}} \paren {y_{K+n}}^{-1} }_{n \in \N}$ an... |
Existence of Logarithm Theorem
Let $b, y \in \R$ such that $b > 1$ and $y > 0$.
Then there exists a unique real $x \in \R$ such that $b^x = y$.
This $x$ is called the logarithm of $y$ to the base $b$. Also see the definition of a (general) logarithm. Proof
We start by establishing a lemma:
Lemma 1
Let $t \in \R$ be suc... |
I’d like to give a simple account of what I call the hierarchy of logical expressivity for fragments of classical propositional logic. The idea is to investigate and classify the expressive power of fragments of the traditional language of propositional logic, with the five familiar logical connectives listed below, by... |
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Say anything you have in your mind! This is for an informal conversation!
Note by Ameya Salankar 5 years, 5 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanat... |
I have system of two equations that describes position of robot end-effector ($X_C, Y_C, Z_C$), in accordance to prismatic joints position ($S_A, S_B$):
$S^2_A - \sqrt3(S_A + S_B)X_C = S^2_B + (S_A - S_B)Y_C$
$X^2_C + Y^2_C + Z^2_C = L^2 - S^2_A + S_A(\sqrt3X_C + Y_C)+M(S^2_A+S_BS_A + S^2_B)$
where M and L are constant... |
Let's take an aqueous solution of a salt $\ce{NaHA}$ with the initial concentration $C$ when added to water. It will completely dissociate according to the eaquation: $\ce{NaHA(s) \rightarrow Na^+ +HA^-}$.
$\ce{HA^-}$ will participate in three equilibria:
$\ce{2HA^- \leftrightarrows H2A +A^{2-}\quad \quad \quad }$ ${K_... |
I have encountered an expression for an inverse Laplace transform, and now I am wondering whether it is correct or maybe there is some error. More likely I don't understand it. so here goes:
$$ g(p)(e^{ap}+\beta)^{-c} \Leftrightarrow \sum_{0\leq n\leq t/a -c}\left(\begin{array}[c] - -c\\ n\end{array}\right)\beta^n f(t-... |
Heat Transfer in Fully-Developed Internal Turbulent Flow From Thermal-FluidsPedia
Line 254: Line 254:
|{{EquationRef|(28)}}
|{{EquationRef|(28)}}
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-
Since the buffer region is also very thin,
+
Since the buffer region is also very thin, <math>1-y/{{r}_{0}}</math> in eq. (5.321) is effectively equal to 1. Defining ... |
How to determine the range of the following function $\frac{x}{1+ |x|}$?
when I calculated it, it was $\mathbb{R}$, but my professor said that the range is ]-1,1[, could anyone explain for me why?
thanks!
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in... |
I am pretty confident that the following limit is $0.5$:
$$\lim_{n\to\infty} \left[\frac{1}{n^{2}} + \frac{2}{n^{2}} + \frac{3}{n^{2}} + \cdots + \frac{n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{1+2+3+ \cdots +n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{n^2+n}{2n^{2}}\right]=\frac{1}{2}$$
However one of the stud... |
I'm going to write $X$ for $(X_1,\ldots,X_n)$ and $x$ for $(x_1,\ldots,x_n)$.
The likelihood function when $X$ is observed to have a certain value $x$ is$$
L_1(\theta) = \Pr(X=x \mid \theta) = E(\Pr(X=x\mid H(X),\theta)) = \sum_h \Pr(H(X)=h\mid\theta)\Pr(X=x\mid H(X)=h).
$$In the very last probability, we don't need to... |
Exam-Style Questions on Graphs Problems on Graphs adapted from questions set in previous Mathematics exams.
1. GCSE Higher
The equation of the line L
1 is \(y = 2 - 5x\).
The equation of the line L
2 is \(3y + 15x + 17 = 0\).
Show that these two lines are parallel.
2. GCSE Higher
Show that line \(5y = 7x - 7\) is perpe... |
I'm doing
Exercise I.11.8 from textbook Analysis I by Amann/Escher.
Show that the identity function and $z \mapsto \overline{z}$ are the only field automorphisms of $\mathbb{C}$ which leave the elements of $\mathbb{R}$ fixed.
Could you please verify if my attempt contains logical gaps/errors?
My attempt:
Let $\phi:\mat... |
Let $\mathcal H^p$, with $p \in [1,\infty)$, be the space of all (continuous-time) martingales $M$ such that $$ \|M\|_{\mathcal H^p} := \mathbb E\left[\sup_t \left| M_t\right|^p\right]^{1/p} < \infty. $$
I want to show that (identifying indistinguishable martingales) $\mathcal H^p$ is complete. Could someone help me co... |
I have the following problem emerged. Let's say we have $l$ finite sets $A_1, A_2, \ldots, A_l$ with cardinality of $n_1, n_2, \ldots\, n_l$, respectively. We know that $| A_i \cap A_j | \le a_{ij}$ (ibvously, $a_{ij}=a_{ji}$).
What could be a lower bound on $\left|\bigcup_{k=1}^l A_k \right|$?
The bound I would be sat... |
Communities (10) Top network posts 139 group by two columns in ggplot2 71 Calculate difference between values in consecutive rows by group 59 How can you read a CSV file in R with different number of columns 39 How to type logarithms in Wolfram|Alpha? 25 Every Function in a Finite Field is a Polynomial Function 21 How ... |
$\log_{b}{b}$ $\,=\,$ $1$
The logarithm of a nonzero positive number to the same quantity is called logarithm of base rule.
Logarithm of any number (nonzero positive number) is equal to one when the same number is taken as a base of the logarithm. There is a reason for this property. When a quantity is split as multipl... |
I am given the series:
$$\sum_{n=1}^{\infty} \frac{\sqrt{n}+\sin(n)}{n^2+5}$$
and I am asked to determine whether it is convergent or not. I know I need to use the comparison test to determine this. I can make a comparison with a harmonic p series ($a_n=\frac{1}{n^p}$ where p > 1, series converges). I argue that as the... |
Using interact
I want to visualize these functions in the complex plane:
$$f_n(z)=\exp(z) - \sum_{k=0}^{n} \frac{z^k}{k!}$$
I tried this code, but it gives me an error.
z,k = var('z,k')@interact def _(n=(1..8)): complex_plot(exp(z)- sum(z^k/factorial(k), k, 0, n), (-5, 5), (-5, 5))
I am new to Sage (I previously used M... |
Searching for just a few words should be enough to get started. If you need to make more complex queries, use the tips below to guide you.
Purchase individual online access for 1 year to this journal.
Impact Factor 2019: 0.808
The journal
Asymptotic Analysis fulfills a twofold function. It aims at publishing original m... |
The angle sum cosine identity is used as a formula to expanded cosine of sum of two angles. For example, $\cos{(A+B)}$, $\cos{(x+y)}$, $\cos{(\alpha+\beta)}$, and so on. Here, you learn how cos of sum of two angles formula is derived in geometric method.
The $\Delta EDF$ is a right triangle and the angle of this triang... |
I suggest to create a tag synonym analysis-and-odes $\to$ ca.classical-analysis-and-odes. The tag analysis-and-odes has only five questions and empty tag info - so it does not seem to have any distinction from ca.classical-analysis-and-odes. Four of the five questions in this tag were asked by t...
I'd like to propose ... |
@egreg It does this "I just need to make use of the standard hyphenation function of LaTeX, except "behind the scenes", without actually typesetting anything." (if not typesetting includes typesetting in a hidden box) it doesn't address the use case that he said he wanted that for
@JosephWright ah yes, unlike the hyphe... |
Searching for just a few words should be enough to get started. If you need to make more complex queries, use the tips below to guide you.
Purchase individual online access for 1 year to this journal.
Impact Factor 2019: 0.808
The journal
Asymptotic Analysis fulfills a twofold function. It aims at publishing original m... |
$\large \log_{b}{(m)} = \dfrac{\log_{d}{(m)}}{\log_{d}{(b)}}$
The base of a logarithmic term can be changed mathematically by expressing it as a quotient of two logarithmic terms which contain another quantity as their base. It is usually called as change of base rule and used as a formula in logarithms.
The change of ... |
I'm trying to choose a group that is hard under the Chosen-Target Computational Diffie-Hellman assumption, according to the definition in this paper, in order to implement the oblivious transfer scheme defined in the top box on page 10(=406).
The (intimidating, to me) CT-CDH assumption is defined as follows (page 7=403... |
Equivalence of Definitions of Separated Sets/Definition 2 implies Definition 1 Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let $A, B \subseteq S$.
Let $U,V \in \tau$ satisfy:
$A \subset U$ and $U \cap B = \empty$ $B \subset V$ and $V \cap A = \empty$
where $\empty$ denotes the empty set.
Then $A^- \cap B... |
As the OP cross-posted this question from Math stackexchange, I have also cross-posted the answer that I wrote there.
The simplest traditional solution to the $d$-dimensional which provides quite good results in 3-dimensions is to use the Halton sequence based on the first three primes numbers (2,3,5). The Halton seque... |
I have a much younger sister who is currently taking calculus as a high school junior. Given that I have a little more mathematical training than anyone else in my family, she sometimes asks me for help with her homework. This is almost invariably a humbling experience.
Last week, she was having trouble with the follow... |
Consider the problem $$ (*)\begin{cases} x''(t)= F(x(t)) \\x(0)=P, x(1)=Q \end{cases}$$ where $P,Q\in\mathbb{R}^3$, $x=(x_1,x_2,x_3)$ and $F=-\nabla U$ for some potential $U:\mathbb{R}^3\rightarrow\mathbb{R}$.
With the assumption of $U$ is upper bounded, I want to show that $(*)$ admits weak solution. This could be don... |
So I have a given lets say $(x+1)^{2x}$ in addition to $\frac{\mathrm dy}{\mathrm dx}a^u=a^u\log(a)u'$. I still have to multiply this by the derivative of the inside function $x+1$ correct?
This is what logarithmic differentiation is for. You start with writing the function as an equation $$y = (x + 1)^{2x},$$ then tak... |
Equivalence of Definitions of Separated Sets/Definition 1 implies Definition 2 Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let $A, B \subseteq S$ satisfy:
$A^- \cap B = A \cap B^- = \empty$ Then there exist $U,V \in \tau$ with: $A \subset U$ and $U \cap B = \empty$ $B \subset V$ and $V \cap A = \empty$ P... |
I know the following type of questions has been asked many times on Math SE. Please let me apologies if my questions seems to be of a repeating nature. For the purpose of my question, we won't consider metric spaces at all.
The notion of open and closed sets between general topological spaces are just called open sets ... |
Distance-Preserving Mapping is Injection of Metric Spaces Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $\phi: M_1 \to M_2$ be a distance-preserving mapping.
Then $\phi$ is an injection.
Proof
Let $a, b \in A_1$ and suppose that $\phi \paren {a} = \phi \paren{b}$.... |
MU First Year Engineering (Semester 1)
Engineering Mechanics May 2016
Engineering Mechanics
May 2016
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 wherever necessary
(1) Assume appro... |
$x$ is an angle of the right angled triangle, $\tan{x}$ is a trigonometric function. A special trigonometric function is formed to represent a value in mathematics as $x$ approaches $\tan^{-1}{3}$.
$\displaystyle \large \lim_{x \,\to\, \tan^{-1}{3}} \normalsize {\dfrac{\tan^2{x}-2\tan{x}-3}{\tan^2{x}-4\tan{x}+3}}$
The ... |
So I have two functions. $f(x) = e^{-x^2+1}$ and $g(x)=\sqrt{x^2-4x+3}$. I am then asked to determine the domain and range of
$a)f∘g,$
$b)g∘f$
I already did part $a)$ and the domain for part $b)$.
For part $a)$, the domain was $(-\infty,1)\cup(3,\infty)$ and the range was $(0,e^2)$.
For part $b$, I figured out that the... |
On a data set I inherited, I calculated the standardized difference between the means of 2 groups,
d, for each of 5 variables. All the data was collected from same set of about 175 people, with 35 in the smaller group. What test might I use to compare the d's? One of the 5 d's is noticeably different from the others, b... |
The question Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$ was recently closed (then reopened and historical-locked). I agree that, by modern standards, the question is not a good one. It is simply a problem statement question, and would rapidly be closed if it were posted today.
However, the question has... |
$$\int_1^2 f(x)\,\mathrm{d}x$$ where $$f(x)=\begin{cases} 1 & \text{if $x$ is rational,} \\ 0 & \text{if $x$ is irrational} \end{cases}$$
How does one interpret this to find the upper and lower sums on a regular partition
$$\Delta x={1\over n}$$
Mathematics Stack Exchange is a question and answer site for people studyi... |
Current browse context:
math.DS
Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Dynamical Systems Title: Polynomial dynamical systems and Korteweg--de Vries equation
(Submitted on 13 May 2016)
Abstract: In this work we explicitly construct polynomial vector fields $\mathcal{L}_k,\;k=0,... |
The exact value of tan of 30 degrees in fraction form is $1/\sqrt{3}$. It can be derived mathematically in three approaches in which two of them are geometric approaches and third one is trigonometric approach. In trigonometry, the tan $30$ degrees value is derived on the basis of values of sin and cos of $30$ degrees.... |
Being a geek and having gone to high school in Los Alamos, I knew about Stanislaw Ulam. At least, I knew a bit about the work he had done during the Manhattan project at Los Alamos. And that he was among the greats of mathematics. But it seemed totally implausible that the person behind that door could really be Stanis... |
$\dfrac{d}{dx}{\, \Big(k.f(x)\Big)} \,=\, k \times \dfrac{d}{dx}{\, f(x)}$
The derivative of product of a constant and a function is equal to the product of constant and the derivative of the function. This property of differentiation is called the constant multiple rule of derivatives.
Let’s take $x$ is a variable, $k... |
ä is in the extended latin block and n is in the basic latin block so there is a transition there, but you would have hoped \setTransitionsForLatin would have not inserted any code at that point as both those blocks are listed as part of the latin block, but apparently not.... — David Carlisle12 secs ago
@egreg you are... |
This is due to the mass-energy equivalence and a phenomenon called binding energy.
Forming a nucleus releases energy because the nucleons are falling into a potential energy well. Due to Einstein's mass energy equivalence this results in the mass of the new nucleus being less than that of the particles that formed it.
... |
$\sin{2\theta} \,=\, 2\sin{\theta}\cos{\theta}$
$\cos{2\theta} \,=\, \cos^2{\theta}-\sin^2{\theta}$
$\tan{2\theta} \,=\, \dfrac{2\tan{\theta}}{1-\tan^2{\theta}}$
$\cot{2\theta} \,=\, \dfrac{\cot^2{\theta}-1}{2\cot{\theta}}$
Learn how to solve easy to difficult mathematics problems of all topics in various methods with ... |
The integration of $\sin^3{x}$ function with respect to $x$ can be calculated in integral calculus in two methods.
The sin and cos functions both have direct relation in integral calculus. If the $sin^3{x}$ function is expressed in terms of both sine and cosine functions. The integral of $\sin^3{x}$ with respect to $x$... |
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... |
In evolutionary psychology, it's a given that genes are selfish, and our genes determine what we are and set the limits on who we can be. (The human brain is very malleable, so those limits aren't always clear in humans but they're nevertheless there.) So we look at human universals like altruism, a sense of justice, h... |
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J/Ψ production and nuclear effects in p-Pb collisions at √sNN=5.02 TeV
(Springer, 2014-02)
Inclusive J/ψ production has been studied with the ALICE detector in p-Pb collisions at the nucleon–nucleon center of mass energy √sNN = 5.02TeV at the CERN LHC. The measurement is performed in... |
Being a geek and having gone to high school in Los Alamos, I knew about Stanislaw Ulam. At least, I knew a bit about the work he had done during the Manhattan project at Los Alamos. And that he was among the greats of mathematics. But it seemed totally implausible that the person behind that door could really be Stanis... |
Let's assume we restrict consideration to symmetric distributions where the mean and variance are finite (so the Cauchy, for example, is excluded from consideration).
Further, I'm going to limit myself initially to continuous unimodal cases, and indeed mostly to 'nice' situations (though I might come back later and dis... |
my new favorite equation a couple months ago, but I neglected to mention an insight that has been revealed to me!
It's Tau Day today! Long live Tau! What does this have to do with my equation? Well, my equation was: [math]\int_{-\infty}^{\infty} e^{-x^2} = \sqrt{\pi}[/math] See Full Post and Comments
It's Tau Day today... |
Equivalence of Definitions of Arborescence Contents Theorem
Let $G = \struct {V, A}$ be a directed graph.
Let $r \in V$.
$G$ is an
arborescence of root $r$ if and only if: For each $v \in V$ there is exactly one directed walk from $r$ to $v$.
$G$ is an
arborescence of root $r$ if and only if: $(1): \quad$ $G$ is an ori... |
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 \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
\(\beta_{0}\) \(=\) \( 1 \) ... |
A fair die is thrown 12,000 times. Use the central limit theorem to find values of $a$ and $b$ such that $$ \mathbb P(1900<S<2200)\approx\int_a^b\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\,\mathrm dx, $$ where $S$ is the total number of sixes thrown.
Right, so the central limit theorem goes as follows:
Let $X_1,X_2,\dots... |
After looking around for a while, having checked the Not-So-Short Introduction to LaTeX2e (2.5.5) and even after having found some sources on the internet, I still could not get this to work. I have been fighting with the problem for a while and it almost seems futile to continue. It must work though, as I have serious... |
[Note: if you are using smartphone or portable device to browser this post, some math formula might not appear properly. To see the math in correct form, scroll down to the bottom and click " View web version"]
After finishing the mini CNC laser engraver, the next thing in my mind is 3D printer. One of the most importa... |
According to the IUPAC Goldbook a
solution is:
A liquid or solid phase containing more than one substance, when for convenience one (or more) substance, which is called the solvent, is treated differently from the other substances, which are called solutes. When, as is often but not necessarily the case, the sum of the... |
When answering such problems, I recommend writing everything know down, then use the appropriate formula to solve for the unknown. For the first part you already did that correctly.
In the laboratory you dissolve $\pu{24.7 g}$ of iron(III) chloride in a volumetric flask in water to a total volume of $\pu{375 ml}$.
\beg... |
"Super-exponential" just means more than exponential, so a function is super-exponential if it grows faster than any exponential function. More formally, this means that it is $\omega(c^n)$ for every constant $c$, i.e., if $\lim_{n\to\infty} f(n)/c(n)=\infty$ for all constants $c$.
Conversely, a function is "sub-expone... |
The distance between the center of the circle to the touching point should be equal to 4 and perpendicular to it.
The equation of the perpendicular line will be:
\begin{equation}g(x)=-1x+b\end{equation}\begin{equation}g(1)=-1*1+b\end{equation}\begin{equation}2=-1*1+b\end{equation}\begin{equation}b=2+1\end{equation}\beg... |
Possible Duplicate: Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$
Hi,
My question is: How can I solve the following integral question? $$\int(\sin ^4 x ) dx$$
Thanks in advance.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It onl... |
D K Choudhury
Articles written in Pramana – Journal of Physics
Volume 60 Issue 3 March 2003 pp 563-567
The exponent λ of the structure function F
2 ∼ −λ is calculated using the solution of the DGLAP equation for gluon at low 2 and as a function of 2 at fixed
Volume 61 Issue 5 November 2003 pp 979-985
The transversity d... |
I'm interested in numerically finding the maximum likelihood estimator of a parameter $\theta$, as well as the confidence interval of this estimator. First I'll describe the method I've been trying, then I'll ask my specific questions about this method.
I perform $N$ binary trials at each possible value of $\theta$ (in... |
Given $2.14\ \mathrm g$ of $\ce{K(s)}$, determine $\Delta H^\circ_\mathrm{f,m}$ and $\Delta U^\circ_\mathrm{f,m}$ for $\ce{K2O}$.
We know:
The calorimeter's constant: $1849\ \mathrm{J\cdot K^{-1}}$ The mass of water inside it: $1450\ \mathrm g$ The change in temperature: $2.62\ \mathrm K$ The end product is $\ce{K2O}$
... |
Searching for just a few words should be enough to get started. If you need to make more complex queries, use the tips below to guide you.
Purchase individual online access for 1 year to this journal.
Impact Factor 2019: 0.808
The journal
Asymptotic Analysis fulfills a twofold function. It aims at publishing original m... |
In my textbook (
Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$):
$$ V_0(y) = \sum_{n=0}^{\infty} C_n\sin{\frac{n\pi}{a}y} \tag{3.31}$$
Here's where things go awry for me.
... how do we actually
determineth... |
I have a problem:
Let $a \in \mathbb{Z}$, $a \geq 3$, and set $\xi= \sum_{n=0}^\infty 10^{-a^{2n}}>0$. Then the inequality
$$\Big|\,\xi - \dfrac{x}{y}\,\Big| \leq \dfrac{1}{y^a}$$ has infinitely many solutions with $x,y \in \mathbb{Z}$, $y>0$ and $\gcd(x,y)=1$.
I would imitate the proof of Dirichlet's theorem as follow... |
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recog... |
I am giving a popular talk on LIGO in 90 minutes and Tristan du Pree has offered me a distraction via Twitter. How do you get distracted if you think about LIGO too much? Yes, by hearing about the LHC: TV:John Oliver gave a totally sensible 20-minute tirade explaining why "scientific study says" stories in the media ar... |
Pfleiderer, C.; Fai\ss{}t, A.; {von L̈ohneysen}, H.; Hayden, S. M.; Lonzarich, G. G.
Title:
Field Dependence of the Specific Heat of Single-Crystalline {{ZrZn2}}
Abstract:
We present measurements of the specific heat C of a single crystal of ZrZn2 in the range 2\textendash{}30K, at magnetic field B up to 14T. For B=0 a... |
I'm trying to maximize a firm's profit given the production function $F(L,K)=L^\alpha K^\beta$ (where $L$ is labor and $K$ is capital) and that $\alpha + \beta \neq 1$.
So, I know that this maximization problem can be written as $\text{max }pF(L,K)-w_1 L-w_2 K$.
Since $pMP_L (L^*,K^*)=w_1$, $p\alpha(L^*)^{\alpha-1}(K^*... |
For an abelian gauge field, the field strength $G_{\mu \nu}$ is gauge-invariant. This means it is a physically observable quantity, e.g. we can build an apparatus to measure electromagnetic field strength.
For a non-abelian gauge field, $G_{\mu \nu}$ transforms non-trivially under infinitesimal gauge transformations $\... |
$\def\A{{\bf A}}\def\B{{\bf B}}\def\C{{\bf C}}\def\R{{\bf R}}\def\D{{\bf D}}\def\f{\phi}$As others have mentioned, if the task is to find the coordinates of $B$ and $C$, this problem is underdetermined.
Let's first consider the triangle with point $A$ located at the origin and point $C$ lying along the positive $x$-axi... |
The mathematical formula for the integration by parts can be derived in integral calculus by the concepts of differential calculus.
$f{(x)}$ and $g{(x)}$ are two functions in terms of $x$. As per the product rule of differentiation, the derivative of the product of the functions $f{(x)}$ and $g{(x)}$ can be written int... |
Partly based on the 438 Fall 2015 lecture material of Professor Boutin.
Contents 1. Introduction
One of the fundamental tools of digital signal processing is the Discrete Time Fourier Transform (DTFT). Unlike the Continuous Time Fourier Transform (CTFT), the DTFT of a signal is periodic with period $ 2\pi $. Although i... |
[1003.0299] The local B-polarization of the CMB: a very sensitive probe of cosmic defects
Authors: Juan Garcia-Bellido, Ruth Durrer, Elisa Fenu, Daniel G. Figueroa, Martin Kunz Abstract: We present a new and especially powerful signature of cosmic strings and other topological or non-topological defects in the polariza... |
Help:Wikitext Examples Contents Basic text formatting
You can format the page using Wikitext special characters.
What it looks like What you type
You can
3 apostrophes will
5 apostrophes will
(Using 4 apostrophes doesn't do anythingspecial --
You can ''italicize text'' by putting 2 apostrophes on '''each''' side. 3 apo... |
Rational Equations
A rational expression is a quotient whose numerator and denominator are polynomials, where the denominator cannot equal zero. For example:
$$\frac{2x^2 +4x -7}{x^2 -3x+8}$$
A rational equation is one that involves only a rational expression. For example:
$$\frac{2x^2 +4x -7}{x^2 -3x+8}=0$$
Solving By... |
Rutherford stated that deflection of alpha particles is due to repulsive positive charge of nucleus. Why can't alpha particles be deflected by attraction of so many electrons revolving around nucleus?
See, what the Geiger-Marsden-Rutherford experiment achieved was the following: by bombarding (with alpha particles) a o... |
Probability Theory: The Logic of Science, by the great E.T. Jaynes, has been in my reading queue for quite some time now. Unfortunately for me it's a dense book after the first few chapters, so I've kind of plateaued around chapter 3 while reading from a bunch of other sources.
I've found that my brain is like boiling ... |
Introduction
In the
Subsymbolic Methods in AI course at NTNU, one of the assignments is to evolve a fully connected feed-forward artificial neural network (ANN) which represents an agent strategy for a Flatland environment. This Flatland environment is a \(10 \times 10\) toroidal grid with cells containing either food,... |
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 $\aleph_2$-saturated ideal over $\omega_1$". [1]
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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 ... |
Yes, this should be solvable and should be doable in a reasonable amount of computation time, using a pretty cool homomorphic cryptosystem.
Here is one approach: the participants jointly pick a random number $y$, publicly commit to $y$, and then they all prove/check in zero knowledge that $y$ is different from their nu... |
This question already has an answer here:
I'm a beginning student of Probability and Statistics and I've been reading the book
Elementary Probability for Applications by Rick Durret.
In this book, he outlines the 4 Axioms of Probability.
For any event $A$, $0 \leq P (A) \leq 1$. If $\Omega $ is the sample space then $P... |
Let $1\leq p < \infty$ and consider a sequence $(x_{n})_{n}\subseteq L^{p}[0,1]$. Show the equivalence of:
$1.$ $x_{n} \xrightarrow{ w} 0$
$2.$ $\sup\limits_{n \in \mathbb N} \vert \vert x_{n}\vert\vert_{p}<\infty $ and $\int_{A}x_{n}(t)dt\xrightarrow{n \to \infty} 0$ for any borel sets $A$ on $[0,1]$.
for $1. \Rightar... |
Today it was told me that wave properties of a particle increase if the temperature decreases. I'm surprised because I have never listened a similar thing, but I think that it's very interesting.
Could you explain me why it happens?
Physics Stack Exchange is a question and answer site for active researchers, academics ... |
I am analyzing the asymptotic runtime of a randomized algorithm in expectation. The algorithm has the following properties:
Given input size $n$, with probability $3/4$ it moves on to solve an instance of size $n-1$ With probability $1/8$ it moves on to solve an instance of size $n-2$ With probability $1/16$ it moves o... |
Equivalence of Definitions of Local Basis Contents Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be an element of $S$.
$\forall U \in \tau: x \in U \implies \exists H \in \mathcal B: H \subseteq U$ Proof $\forall U \in \tau: x \in U \implies \exists H \in \mathcal B: H \subseteq U$
Let $N$ be a ne... |
As with most people that have studied mathematics beyond high school, I first encountered implicit differentiation in my initial semester of calculus. It seemed like magic then. In the intervening years, I have seen the idea and the underlying theory pop up again a few times, and it still seems like some kind of magic.... |
State functions such as $G$ only depend on the state of the system and are not dependent on the "path" that took the system to that state (which would be the case for work, for example, which is not a state function.
We know that:
$$G=V\mathrm{d}p-S\mathrm{d}T$$ So... $$G=\left(\frac{\partial G}{\partial p}\right)_T\ma... |
I am wondering to what degree we can define an RSA variant, with a security argument that it is as safe as regular RSA with a given modulus size $m$ (e.g. $m=2048$), in which the
public key has a compact representation of $k \ll m$ bits.
We can fix the public exponent to our favorite customary value, e.g. $e=2^{16}+1$ ... |
I want to store a subset of $\{1,2,\dots,n\}$ in a data structure such that the total space used is $O(n)$ bits and accessing a particular element can be done in $O(1)$ time. I have tried binary search tree and array data structures but they are too slow.
In short I need a data structure with $O\big(\frac {n } { \log \... |
Let $n,m \in \mathbb{N}$ $$n=\prod_{i=1}^{r}p_{i}^{a_i}$$ where $p_i$ are prime factors and $f$ , $g$ and $h$ are the functions $$f(n,m)=\sum_{j=1}^{n}j^m$$ And $$g(n)=\sum_{i=1}^{r}a_i.p_i$$ If we put $m=1,n=21$ then $$g(f(21,1))=g(231)=21.$$
21 is only number satisfy $g(f(n,1))=n$.
Now let
$$h(m) = \sum_{g(f(n,m))=n}... |
Combination Theorem for Sequences/Normed Division Ring/Product Rule/Proof 2 Theorem
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring.
Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$.
$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ $\displaystyle \lim_{n \mathop \to \infty} y_n = m$
Then:... |
So here in the picture \$e^{jwt}\$ is the input to the system and \$h(t)\$ is the impulse response. So, by convolution integral shouldn't the response be \$h(t-T)e^{jwT} dT\$? But here it is \$h(T)e^{jw(t-T)}dT\$. What am I missing here?
Nothing. Just do a variable change \$\tau'=t-\tau\$ and you'll get your integral.
... |
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