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The standard proof, apparently due to Dedekind, that algebraic numbers form a field is quick and slick; it uses the fact that $[F(\alpha) : F]$ is finite iff $\alpha$ is algebraic, and entirely avoids the (to me, essential) issue that algebraic numbers are roots of some (minimal) polynomial. This seems to be because fi... |
This table is, indeed, calculated by assuming that a ship can produce constant acceleration away from its origin, instantaneously pivot \$180^\circ\$ at the midpoint of its journey, and constantly decelerate for the second half of its journey. The greatest veloticy thus attained (w.r.t. origin) in the table above is ro... |
exponential equation solve problem
b=4*3^(2*x-1)==5*4^(x+2)show(b)solve(b,x)
The Solution is:
[4^(x + 2) == 4/5*3^(2*x - 1)]
Should the solve alg. solve for x?
Well...
sage: b.solve(x)[0].log().log_expand().solve(x)[x == 1/2*(log(3) + 4*log(2) - log(4/5))/(log(3) - log(2))]
<Swing>"Who could ask for anything more ?"
</... |
Difference between revisions of "Group cohomology of dihedral group:D8"
(→Over the integers)
(→Over the integers)
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The homology groups over the integers are given as follows:
The homology groups over the integers are given as follows:
−
<math>H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} (\math... |
Compressed Sensing (CS) is a technique that allows the reconstruction of signals starting from a limited number of linear
measurements that is potentially much smaller than the number of Nyquist-rate samples. The possibility of such sub-Nyquist representation hinges on an assumption on the considered class of signals, ... |
Table of Contents
Fundamental Groups under Homeomorphisms on Topological Spaces
We now look at a very important theorem regarding the fundamental groups of topological spaces.
Theorem 1: Let $X$ and $Y$ be path-connected topological spaces. If $X$ and $Y$ are homotopically equivalent then $\pi_1(X, x) \cong \pi_1(Y, y)... |
Continued fractions provide a representation of numbers which is, in a sense, generic and canonical. It does not depend on an arbitrary choice of a base. Such a representation should be the best in a sense. In this section we quantify this naive idea.
A rational number \(a/b\) is referred to as a "good" approximation t... |
Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.
I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’... |
Learning Objectives
In this section students will:
Identify the degree and leading coefficient of polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL to multiply binomials. Perform operations with polynomia ls of several variables.
Earl is building a doghouse, whose front is in the shape of a squa... |
I'm trying to prove what seems an elementary general topology exercise (I'm trying to prove it in order to use it in a basic complex analysis course) It's the following property:
Let $(X,d)$ be a metric space, let $G\subset X$ be an open but not closed subset of $X$ and $x\in G$. Denoting, for each $r>0$, the open ball... |
I talking about functions of the form ||X||^p, when p>1 for different values of p.
I know these are all convex functions, but I don't know how to graph them.
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... |
This question is somehow related to my question at https://math.stackexchange.com/questions/683915/derived-pseudo-functor and to the question here: A homotopy commutative diagram that cannot be strictified .
Consider a model category $M$. Let us assume that, for all small category $S$, we can endow the category $Fun (S... |
This is a heuristic explanation of Witten's statement, without going into the subtleties of axiomatic quantum field theory issues, such as vacuum polarization or renormalization.
A particle is characterized by a definite momentum plus possible other quantum numbers. Thus, one particle states are by definition states wi... |
Higher Order Homogenous Differential Equations Real, Distinct Roots of The Characteristic Equation
Consider the following $n^{\mathrm{th}}$ order linear homogenous differential equation with the constant coefficients $a_0, a_1, ..., a_n \in \mathbb{R}$:(1)
Recall from the Higher Order Homogenous Differential Equations ... |
When you studied fractions, you had lots of different ways to think about them. But the first way, and the one we keep coming back to, is to think of a fraction as the answer to a division problem.
Example \(\PageIndex{1}\):
Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per child. We writ... |
Is there a standard way of writing
$a$ is divisible by $b$ in mathematical notation?
From what I've search it seems that writing $a \equiv 0 \pmod b$ is one way? But also you can write $b \mid a$ as well (the middle character is a pipe)? And sometimes that pipe is replaced by $3$ vertical dots?
Or is there a way of wri... |
Search
Now showing items 1-1 of 1
Higher harmonic flow coefficients of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
(Springer, 2016-09)
The elliptic, triangular, quadrangular and pentagonal anisotropic flow coefficients for $\pi^{\pm}$, $\mathrm{K}^{\pm}$ and p+$\overline{\mathrm{p}}$ in Pb-... |
The Method of Undetermined Coefficients
We will now look at a method for solving second order linear nonhomogenous differential equations, for $a, b, c \in \mathbb{R}$, in the form:(1)
Note that the corresponding second order linear homogenous differential equation $a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = 0$ has c... |
Words Over a Set
Definition: Let $A = \{ a, b, ... \}$ be a nonempty set. The Inverse Set of $A$ is the set $A^{-} = \{ a^{-1}, b^{-1}, ... \}$. A Word over $A \cup A^{-}$ is a finite sequence $w$ of elements in $A \cup A^{-1}$. The Empty Word is the empty sequence $()$. We are not assuming a group context here. If $A$... |
Taiwanese Journal of Mathematics Taiwanese J. Math. Volume 19, Number 2 (2015), 381-396. INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS Abstract
In this paper, we deal with the existence of infinitely many solutions for a class of sublinear Schrödinger equation $$ \left\{ \begin{array}{ll} -\t... |
Retract Subspaces of a Topological Space
Definition: Let $X$ be a topological space and let $A \subset X$ be a topological subspace. Then $A$ is said to be a Retract of $X$ if there exists a continuous function $r : X \to A$ called a Retraction Map such that $r \circ \mathrm{in} = \mathrm{id}_A$. Here, $\mathrm{in} : A... |
One disadvantage of the fact that you have posted 5 identical answers (1, 2, 3, 4, 5) is that if other users have some comments about the website you created, they will post them in all these place. If you have some place online where you would like to receive feedback, you should probably also add link to that. — Mart... |
In my syllabus we only study about rate laws under the topic "Rates of reactions", but as I was interested in it, I started to search about that topic. And then I found the Arrhenius equation$$\ln k ...
I need to calculate the half life for a first order reaction for which I need the rate constant $k$. From literature ... |
Answer
$s=\dfrac{\pi}{4}$
Work Step by Step
RECALL: $\frac{\pi}{4}$ is a special angle and $\cos{(\frac{\pi}{4})}=\dfrac{\sqrt2}{2}$ Thus, if $\cos{s} = \frac{\sqrt2}{2}$, then $s=\dfrac{\pi}{4}$.
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll hav... |
Analysis of the Minimal Compliance Problem and Related Filters Andreas Thalhammer Nov. 25, 2014, 3:30 p.m. S2 059
In this talk we consider the topology optimization of elastic continua. For this, we derive the so-called \textit{Minimal Compliance Problem:}
\begin{align*}
\ell(\mathbf{u}(\rho)) &\to \min_{\rho\in L_\inf... |
Literature on Carbon Nanotube Research
I have hijacked this page to write down my views on the literature on Carbon Nanotube (CNT) growths and processing, a procedure that should give us the cable/ribbon we desire for the space elevator. I will try to put as much information as possible here. If anyone has something to... |
Most of us think that counting is as easy as 1, 2, 3... When counting objects, one needs to be careful to not count an object more than once or miss an object. In this section, we will explore some ideas behind counting.
The Multiplication Principle
If a process can be broken down into two steps, performed in order, wi... |
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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 ... |
No, they're mostly notational variations. There are different connotations to the different notations, and different notations are common in different fields where they can mean quite different things. Also, sometimes they are used in a particular context for different (but usually related) things. You'll, of course, h... |
The weight 2 Bianchi modular forms are particularly important in regard to their conjectural connections with abelian varieties of $\textrm{GL}_2$-type. In the weight 2 case, we have $F: \mathcal{H}_3 \rightarrow \mathbb{C}^3$ and $$(F |_k\gamma)(z)=\dfrac{1}{|r|^2+|s|^2} \begin{pmatrix} \bar{r}^2 & 2\bar{r}s & s^2 \\ ... |
For exercises 1 - 15, find \(f′(x)\) for each function.
1) \(f(x)=x^2e^x\)
Answer: \(f'(x) = 2xe^x+x^2e^x\)
2) \(f(x)=\dfrac{e^{−x}}{x}\)
3) \(f(x)=e^{x^3\ln x}\)
Answer: \(f'(x) = (3x^2\ln x+x^2)e^{x^3}\ln x\)
4) \(f(x)=\sqrt{e^{2x}+2x}\)
5) \(f(x)=\dfrac{e^x−e^{−x}}{e^x+e^{−x}}\)
Answer: \(f'(x) = \dfrac{4}{(e^x+e^{−... |
Note: This question was asked in stats.stackexchange.com and math.stackexchange.com, with expired bounties on both sites.
Given a sequence of iid random variables $X_i$ (without loss of generality from $U(0,1)$), an integer $k \ge 1$ and some $p \in (0,1)$, construct the sequence of random vectors $Z^{(j)}$, $j=0,1,...... |
Let me answer the second query.
We know $$\mathbf{ B}=\text{curl}\;\mathbf{ A} $$ where $\bf B$ is the magnetic field & $\bf A$ is vector-potential.
Now, using Maxwell's equation, we get $$\text{curl}\;(\text{curl}\; \mathbf A)= \mu_0 \mathbf J . $$ By cracking a bit-algebra, we get, $$-\frac{\partial^2 A_x}{\partial x... |
Saying that warm air "holds" more moisture is technically incorrect, but is a common colloquialism. Let's break it down to the technicalities.
Let's consider a glass of water with a vacuum (no air) above it. What will happen? The molecules that are at the top most layer of the water will evaporate. At what rate will th... |
Prove that the function $\sqrt x$ is uniformly continuous on $\{x\in \mathbb{R} | x \ge 0\}$.
To show uniformly continuity I must show for a given $\epsilon > 0$ there exists a $\delta>0$ such that for all $x_1, x_2 \in \mathbb{R}$ we have $|x_1 - x_2| < \delta$ implies that $|f(x_1) - f(x_2)|< \epsilon.$
What I did wa... |
Inaccessible cardinal Inaccessible cardinals are the traditional entry-point to the large cardinal hierarchy, although weaker notions such as the worldly cardinals can still be viewed as large cardinals.
A cardinal $\kappa$ being inaccessible implies the following:
$V_\kappa$ is a model of ZFC and so inaccessible cardi... |
Now let \(\theta \) be any angle. We say that \(\theta \) is in
standard position if its initial side is the positive \(x\)-axis and its vertex is the origin \((0,0) \). Pick any point \((x,y) \) on the terminal side of \(\theta \) a distance \(r>0 \) from the origin (see Figure 1.4.3(c)). (Note that \(r = \sqrt{ x^2 +... |
I am bit confused as to if both the things are the same since it seems that people refer to both as being given by the ``Liouville action",
\(\frac{1}{4\pi}\int d^2z ( \vert \partial \phi \vert ^2 + \mu e^{\phi } ) + bounday-terms \)
- Is the above action (which I would have thought is for the Liouville CFT) the same a... |
I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind.
I found the integral expression below on Gradshteyn and Ryzhik's book 7th edition, section 6.677, number 6: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\rig... |
Let us show that $a_n=\left(1+\frac{1}{n}\right)^n$ for $n\geq 1$ gives an increasing sequence bounded by $3$.
Increasing. The product of $k$ positive numbers can always be written as the $k$-th power of their geometric mean. In particular$$ 1\cdot\left(1+\frac{1}{n}\right)^n = \text{GM}\big(1,\underbrace{1+\tfrac{1}{n... |
Is the particle in a box under harmonic driving electric field solvable analytically?
Here is the Schrodinger equation:
$$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$
where the potential $V(x)$ is
$$ V(x)= \begin{array}{cc} \Big\{ & \begin{arr... |
Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_Y$. There are no dependencies between these two pairs. Now I want to know the following expected va... |
I needed to compute $\sin 18^{\circ}$.
Now, these two relations hold for every $x$:
$\cos 5x=16\cos^5x-20\cos^3x+5\cos x$ $\sin5x=16\sin^5x-20\sin^3x+5\sin x$, which can be easily proved using the multiple angle formulae.
Now, one thing to observe is : $\sin5(18^{\circ})=1$ and $\cos5(18^{\circ})=0$
So, $$\begin{align}... |
Finite temperature is introduced in the Ads Space by inserting a black hole. In the Ads-CFT correspondence, the Wilson loop is at $u \rightarrow \infty$. But the black hole horizon itself would be at $u \rightarrow u_0$, i.e. a finite u, implying that the black hole mass is finite.
For reference, let me specify the met... |
How can I as a trusted user of a middleman company (such as PhishTank) verify whether a phishing site is valid if the scam listens only on a unique referrer link(randomly created) and is blocking any other access methods?
To throw a threat scenario into scene.
An attacker sent an email to a local bank officer, the emai... |
Tangent Lines at Points
Consider a curve represented by the function $f$, and suppose that we want to find the slope of a line tangent to the point $P(a, f(a))$. To calculate this slope, suppose we take the point $Q(x, f(x))$. We can easily create a secant line from $P$ to $Q$ from which we use basic algebra to calcula... |
Table of Contents
The Method of Undetermined Coefficients Examples 1
Recall from The Method of Undetermined Coefficients page that if we have a second order linear nonhomogeneous differential equation with constant coefficients of the form $a \frac{d^2y}{dt^2} + b \frac{dy}{dt} + cy = g(t)$ where $a, b, c \in \mathbb{R... |
I am looking for some graph theory concepts and definitions around embedding a DAG into another DAG. I could only find a few lines on Wikipedia around this so I wonder if someone can help me find good references for the important concepts and definitions.
For example, assume I have a DAG $G=(E,V)$ where there are two t... |
I apologize that this is perhaps not adequate for mathoverflow but I have struggled with this for days now and become desperate...
The reduced K-group $\tilde{K}(S^0)$ of the zero sphere is the ring $\mathbb{Z}$ as being the kernel of the ring morphism $K(S^0)\to K(x_0)$. The ring structure on $K(S^0)$ and $K(x_0)$ com... |
I was told that using gradient methods for Gaussian mixture models may end up with Dirac delta function(s). I hadn't thought of this problem before, but when I verify this, it does seem to be a problem.
For example, let us consider a mixture of 2 Gaussians, and data points $x_1, x_2, \cdots, x_m$ ($m\gg$ 2). The follow... |
The definition of incompressible is often unclear and changes depending on which community uses it. So let's look at some common definitions:
Constant density
This means the density is constant everywhere in space and time. So:$$\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + \vec{u}\cdot\nabla{\rho} = 0$$Because... |
I need your help for my task.
I need to calibrate to the market data for Hull White model for Zero Coupon Bond Price. I refer to John Hull and Alan White paper. I want to ask you a few questions and correct me if I'm wrong in my steps.
We need to know the spot rate $r$ at a certain time $t$ and yield curve at a certain... |
On the last question, I am not sure how good you are at the representation theory, but the following fact is true: take so(d,2) (we need so(3,2) for this work), use the conformal base, i.e. Lorentz generators $L_{ab}$, translations $P_a$, conformal boosts $K_a$ and dilatation $D$, $a,b=1..d$. $P$ and $K$ behave as rais... |
Brouwer's Fixed Point Theorem
Recall from one of the results on The Induced Mapping from the Fundamental Groups of Two Topological Spaces page that if $S^1$ is not a retract of $D^2$. We will use this result to prove the famous Brouwer's fixed point theorem.
Theorem 1 (Brouwer's Fixed Point Theorem): Every continuous f... |
A discrete memoryless source
W has words $w_1,w_2,w_3,w_4,w_5,w_6$ that occur with probablilities $0.05,0.05,0.15,0.2,0.25,0.3$ respectivley.
Does there exist a compact instantaneous binary encoding for this source with word lengths $2, 2, 4, 4, 5$ and $5$?
Shannon's Noiseless coding theroem that says a compact encodin... |
Notice:
If you happen to see a question you know the answer to, please do chime in and help your fellow community members. We encourage our fourm members to be more involved, jump in and help out your fellow researchers with their questions. GATK forum is a community forum and helping each other with using GATK tools a... |
If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ?
We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not be rational:
Denote $x=\dfrac{b}{a},(a,b)=1,x^x=\dfrac{d}{c},(c,d)=1,$ $$\left(\dfrac{b}{a}\right)^\dfrac{b}{a}=\dfrac{d}{... |
Moving-average smoothing
ma computes a simple moving average smoother of a given time series.
Keywords ts Usage
ma(x, order, centre = TRUE)
Arguments x
Univariate time series
order
Order of moving average smoother
centre
If TRUE, then the moving average is centred for even orders.
Details
The moving average smoother av... |
In my post Trigonometry Yoga, I discussed how defining sine and cosine as lengths of segments in a unit circle helps develop intuition for these functions.
I learned the circle definitions of sine and cosine in my junior year of high school, in the class that would now be called pre-calculus (it was called “Trig Senior... |
Assume I have these two elliptic curves: \begin{align*} E:Y^2&=X^3+b_2X^2+b_4X+b_6\\ E':Y^2&=X^3+gb_2X^2+g^2b_4X+g^3b_6, \end{align*} over $\mathbb{F}_q$, where $g$ is not a square in $\mathbb{F}_q$, and $\mathbb{F}_q$ does not have characteristic $2$. I know that $\#E(\mathbb{F}_q)=q+1-t$ and am asked to prove that $\... |
I’m not sure to whom the image or the idea is due. Please comment if you have information. (See comments below for current information.)
The rules will naturally generalize those in Connect-Four. Namely, starting from an empty board, the players take turns placing their coins into the $\omega\times 4$ grid. When a coin... |
Reduction of Order on Second Order Linear Homogeneous Differential Equations Examples 1
Recall from the Reduction of Order on Second Order Linear Homogenous Differential Equations page that if we have a second order linear homogeneous differential equation $\frac{d^2y}{dt^2} + p(t) \frac{dy}{dt} + q(t) y = 0$ and if $y... |
I had a problem in my booked i tried to prove. Here is the problem
"Let $x_1,\ldots,x_n$ be different real numbers and $y_1,\ldots,y_n,s_1,\ldots,s_n$ some real numbers. Prove that there exists a polynomial $p(x)$ of a degree less than $2n$ such that $p(x_i)=y_i$ and $p'(x_i)=s_i$ for every $i=1,2,\ldots,n.$"
Here is m... |
An exponential sum is an expression of the form \[ \sum_{n=1}^N e^{2 \pi i f(n)},\] where \( f \) is a real-valued function defined on the positive integers. Such sums are used in the solution of various problems in number theory; in this article we will just play around with a few examples, draw their graphs and try t... |
So I'd like some help w/ this question.
Given 3 assets with means, variances, and correlation:
Two portfolios are created (A and B), each with the three assets above with weights ($w_n$) as follows:
Portfolio A: $w_1=0.2$, $w_2=0$, $w_3=0.8$
Portfolio B: $w_1=0.4$, $w_2=0.1$, $w_3=0.5$.
The assets' correlation are
$\rh... |
Invertibility of a Linear Map
Invertibility of a Linear Map
Definition: If $T \in \mathcal L (V, W)$ then the linear map $T$ is said to be Invertible if $\exists S \in \mathcal L (W, V)$ such that $ST = I_V$ and $TS = I_W$. The linear map $S$ is said to be the Inverse Linear Map of $T$ which we denote by $S = T^{-1}$.
... |
The issue can be characterized as a confusion of prior and posterior probabilityor maybe as the dissatisfaction of not knowing the joint distribution of certain random variables.
Conditioning
As an introductory example,we consider a model for the experiment of drawing, without replacement,two balls from an urn with $n$... |
Basic Theorems Regarding the Boundary of a Set in a Topological Space
Recall from The Boundary of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then a point $x \in X$ is called a boundary point of $A$ if $x$ is contained in the closure of $A$ but not in the interior of... |
Complex Numbers Examples 1
Recall from the Complex Numbers page that numbers in the form $z = a + bi$ where $a, b \in \mathbb{R}$ and $i = \sqrt{-1}$ are called complex numbers and the set of all complex numbers is denoted by $\mathbb{C}$. We will now look at some examples regarding complex numbers.
Example 1 Graph the... |
Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.
I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’... |
Recall that a differential equation is an equation (has an equal sign) that involves derivatives. Just as biologists have a classification system for life, mathematicians have a classification system for differential equations. We can place all differential equation into two types: ordinary differential equation and pa... |
I'm taking a stochastic processes class, and we looked at the example of Gambler's ruin with infinite target, i.e. the gambler stops when he reaches 0 fortune or N, in the limit of N going to infinity.
For a finite target, in the case of $p=q$, the probability to ruin and expected time to ruin for starting fortune $a$ ... |
For an elliptic curve over Q that is defined with large coefficients, it can take mathematical software (such as Sage) a long to time calculate the analytic rank. However, it seems to quickly know if the rank is even or odd.
I would like to understand how they determine this so quickly.
This is hinted at in a PlanetMat... |
Dirac's Theorem
Theorem 1 (Dirac's Theorem): If $G = (V(G), E(G))$ is connected graph on n-vertices so that for every $x, y \in V(G)$, where $x \neq y$, and $\deg (x) + \deg (y) ≥ n$ for all $x, y \in V(G)$, then $G$ is a Hamiltonian graph.
Let's verify Dirac's theorem by testing to see if the following graph is Hamilt... |
The Law of Cosines is presented as a geometric result that relates the parts of a triangle:
While true, there’s a deeper principle at work.
The Law of Interactions: The whole is based on the parts and the interaction between them.
The wording “Law of Cosines” gets you thinking about the mechanics of the formula, not wh... |
2018-08-25 06:58
Recent developments of the CERN RD50 collaboration / Menichelli, David (U. Florence (main) ; INFN, Florence)/CERN RD50 The objective of the RD50 collaboration is to develop radiation hard semiconductor detectors for very high luminosity colliders, particularly to face the requirements of the possible u... |
Preliminary Definitions for The Theory of First Order ODEs
Preliminary Definitions for The Theory of First Order ODEs
Before we move on to some of the theory regarding first order ordinary differential equations we will need to state some important definitions from real analysis.
Pointwise and Uniform Convergence of Se... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
A) Atom is indivisible
B) Gases combine in a simple ratio
C) There is no influence of gravity on the molecules of a gas
D) None of the aboveView Solution
A) There are intermolecular attractions
B) Molecules have considerable volume
C) No intermolecular attractions
D) The velocity of molecules decreases after each colli... |
Spatial resolution Geometric effects
When working with a transmission electron microscope (TEM) in scanning (STEM) or focused probe mode, the spatial resolution depends of several effects. For probes greater than ~2 nm and thicker samples (greater than ~ 75 nm), you can approximate the resolution with simple geometric ... |
Covering Maps are Open Maps
Recall from the Covering Spaces page that if $X$ is a topological space then a covering space of $X$ is a pair $(\tilde{X}, p)$ where $\tilde{X}$ is a path connected and locally path connected topological space and $p : \tilde{X} \to X$ is a continuous map such that for every $x \in X$ there... |
Table of Contents
The Interior Points of Sets in a Topological Space
Recall from the The Open Neighbourhoods of Points in a Topological Space page that if $(X, \tau)$ is a topological space and $x \in X$ then an open neighbourhood of $x$ is any open set $U$ ($U \in \tau$) such that $x \in U$.
Given a subset $A \subsete... |
Every day one sees politicians on TV assuring us that nuclear deterrence works because there no nuclear weapon has been exploded in anger since 1945. They clearly have no understanding of statistics.
With a few plausible assumptions, we can easily calculate that the time until the next bomb explodes could be as little ... |
Hello one and all! Is anyone here familiar with planar prolate spheroidal coordinates? I am reading a book on dynamics and the author states If we introduce planar prolate spheroidal coordinates $(R, \sigma)$ based on the distance parameter $b$, then, in terms of the Cartesian coordinates $(x, z)$ and also of the plane... |
Table of Contents
Second Countability under Homeomorphisms on Topological Spaces
Recall from the Homeomorphisms on Topological Spaces page that if $X$ and $Y$ are topological spaces then a bijective map $f : X \to Y$ is said to be a homeomorphism if it is continuous and open.
Furthermore, if such a homeomorphism exists... |
I am looking for a nice and readable description of how to implement BDT model: $d log(r(t)) = [\theta(t)-\frac{\sigma'(t)}{\sigma(t)}log(r(t))]dt + \sigma(t) dW$.
I assume I already have steady-state IR curve $r^*(t)$ and volatility curve $\sigma^*(t)$.
It makes no difference whether it would be binomial tree or Monte... |
Basis of a Vector Space
We will now look at a new definition regarding vector spaces.
Definition: A set of vectors $\{ v_1, v_2, ..., v_n \}$ is said to be a Basis of the $\mathbb{F}$-vector space $V$ if both $V = \mathrm{span} (v_1, v_2, ..., v_n)$ and $\{v_1, v_2, ..., v_n \}$ is a linearly independent set.
From the ... |
I've been searching a lot for relevant answers to my question. However, I was unable to find a problem formulation with satisfactory answers that would help for my problem.
In a nutshell, I would like to find an initial feasible basis, $\mathcal{B}$, to start my simplex algorithm. And I would like to find this initial ... |
Uplifting cardinals Uplifting cardinals were introduced by Hamkins and Johnstone in [1], from which some of this text is adapted.
An inaccessible cardinal $\kappa$ is
uplifting if and only if for every ordinal $\theta$ it is $\theta$-uplifting, meaning that there is an inaccessible $\gamma>\theta$ such that $V_\kappa\p... |
Defining how connected a graph is, is sometimes difficult. We will now define connectivity of a graph in terms of what are called vertex cutsets and edge cutsets.
Vertex Cutsets
Definition: For a connected graph $G = (V(G), E(G))$, a Vertex Cutset is a subset $W$ of the vertex set $W \subseteq V(G)$ if and only if the ... |
Let $X$ be a continuous random variable and $Q_x$ is the associated quantile function. Show that expected shortfall $ES_X[p]$ at the confidence level $p$ which is defined as
$$ES_X[p]=\Bbb E[X|X\leq Q_x(1-p)]$$ has the representation $$ES_X[p]=\frac{1}{1-p}\int_0^{1-p} Q_x(a)da.$$
Can some one give me a hint for this?
... |
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Can someone please help me out with the following question?
Q. A simple harmonic oscillator, of mass m and natural frequency w_0, experiences an oscillating driving force f(t) = macos(wt). Therefore its equation of motion is:
[tex]\frac{{d^2 x}}{{dt^2 }} + \omega _0 ^2 x = a\cos \left( {\omega t} \right)[/tex]
Gi... |
Compactness of Finite Sets in a Topological Space
Recall from the Compactness of Sets in a Topological Space page that if $X$ is a topological space then a set $A \subseteq X$ is said to be compact in $X$ if every open cover of $X$ has a finite subcover.
We will now look at a nice theorem that says that any finite set ... |
\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\math... |
To begin, we need to find distances. Starting with the Pythagorean Theorem, which relates the sides of a right triangle, we can find the distance between two points.
Definition: Pythagorean Theorem
The Pythagorean Theorem states that the sum of the squares of the legs of a right triangle will equal the square of the hy... |
If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is
space homogeneous, you have that the price of a European vanilla option is itself a space-homogeneous function of degree one. You can then appeal to Euler theorem to get the relationship you need.
More specifically, ... |
Path Connectivity of Connected Topological Spaces
Recall from the Path Connected Topological Spaces page that a topological space $X$ is said to be path connected if for each pair of points $x, y \in X$ there exists a continuous function $\alpha : [0, 1] \to X$, call a path, such that $\alpha(0) = x$ and $\alpha(1) = y... |
Before introducing the Dirac equation, it was difficult to explain the behaviour of the particles as the particles with higher velocities were not studied. But Dirac equation introduced four new components to the wave. These four components were divided into two energy states: positive and negative. Both energy states ... |
Currently going through a video on Counting Minimum Cuts by Tim Roughgarden. $(A_{i},B_{i}) = \big((A_{1},B_{1}), ..., (A_{t},B_{t})\big) \forall i \in \Bbb{R}$ $P\big((A_{i},B_{i})\big) \geq \frac{1}{\begin{pmatrix} n \\ 2 \end{pmatrix}} = p$, which I interpret as the lower bound on the probability of having at least ... |
This is something I haven't seen online yet, indicator functions with values in a finite field. Probably for a good reason, but I would like to know why, and if there are still things that can be said. For instance what can we say of the relationship (if any) between the support of the convolution of two indicator func... |
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