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"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
The short rate in the Ho-Lee model is given by :
$$dr_t=\left( \frac{df(0,t)}{dt} +\sigma^2t\right)dt + \sigma dW_t$$
I'm trying to find the bond dynamics given by :
$$dP(t,T)/P(t,T)=r_tdt-\sigma(T-t)dW_t$$
I started from :
$$P(t,T)=E_t[e^{-\int_t^T r_sds}]$$
and I applied Itô to the function $P(t,T)=\phi(t,r)$:
$$d\ph... |
Answer
a. Natural: $\sqrt{100}$ b. Whole: $0,\sqrt{100}$ c. Integer: $-9,0,\sqrt{100}$ d. Rational: $-9,-\displaystyle \frac{4}{5},0.25,9.2,\sqrt{100}$ e. Irrational: $\sqrt{3}$ f. Real: all of them
Work Step by Step
The set of natural numbers is { 1, 2, 3, 4, 5, ... } . The set of whole numbers is { 0, 1, 2, 3, 4, 5, ... |
Possible Duplicate: Why does 1/x diverge?
I'm a math tutor. This is a high school level problem. I'm unable to solve this.
What is the value of:
$\lim\limits_{n \to \infty}\sum\limits_{k=1}^n \frac{1}{k}$
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in... |
Please illustrate that a bond with maturity N years that has coupon equal to its yield is associated with the conversion factor of 1.
I do this by writing out $$\frac1{100} \left( \sum_{t=1}^N \left[ \frac{100 (0.06)}{1.06^t} \right]+\frac{100}{1.06^N} \right)$$ but I do not get that this = 1.
I use the formula:
$$\sum... |
Regularity under sharp anisotropic general growth conditions
1.
Dipartimento di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy, Italy
sharpassumptions on the exponents $p_{i}$ in terms of $\overline{p} $: the * Sobolev conjugate exponentof $\overline{p}$; i.e., $\overline{p} $ ... |
A vertical asymptote is a place where the function becomes infinite, typically because the formula for the function has a denominator that becomes zero. For example, the reciprocal function $f(x)=1/x$ has a vertical asymptote at $x=0$, and the function $\tan x$ has a vertical asymptote at $x=\pi/2$ (and also at $x=-\pi... |
We still have not answered one of our first questions about the steepness of a surface: starting at a point on a surface given by $f(x,y)$, and walking in a particular direction, how steep is the surface? We are now ready to answer the question.
We already know roughly what has to be done: as shown in figure 16.3.1, we... |
To familiarize myself with concepts from my system modeling class, I have posed myself the following problem, which I have been struggling with for a good while now. I am an absolute physics beginner, so it is likely I fundamentally misunderstood some things about pressure etc. The Problem
There are two water tanks wit... |
The point is the following:
Delta, $\Delta$, is defined as $\frac{\partial C}{\partial S}$, where $C$ is the value of the call option, and $S$ is the price of the underlying asset.
So, given that the value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is
$$C = N(d_... |
Define
$$y(x,s) = \int_0^{\infty} dt \, Y(x,t) \, e^{-s t}$$
Then, integrating by parts:
$$\int_0^{\infty} dt \, Y_t(x,t) \, e^{-s t} = -Y(x,0) + s y(x,s)$$
$$\int_0^{\infty} dt \, Y_{tt}(x,t) \, e^{-s t} = -Y_t(x,0) + s Y(x,0) + s^2 y(x,s)$$
Then using the initial conditions $Y(x,0)=Y_y(x,0)=0$, the PDE becomes the fo... |
One of the challenges of being a math teacher is getting all the fractions and square roots into your documents. The challenge is especially large on the web. Most websites do not have a way for teachers to enter math symbols and we have to make due with calculator expressions.
Year ago a colleague, Gayle Taylor, intro... |
"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
I like the Connectedness argument, which follows straight from the axioms of a topology. A topological space $\left(\mathbf{X},\,\mathcal{T}\right)$ is connected iff $\mathbf{X}$ and $\emptyset$ are the only members of $\mathcal{T}$ which are both open and closed at once. $\mathbf{A} \subset \mathbf{X}$ is both open an... |
I was following the textbook by David Mackay:
Information theory inference and learning algorithms.
I have question on asymptotic equiparition' principle:
For an ensemble of $N$ $i.i.d$ random variables $X^N=(X_1,X_2....X_N),$ with $N$ sufficiently large, the outcome $x=(x_1,x_2...x_N)$ is almost certain to belong to a... |
I have found out the answer.
When you do a measurement (one measurement) you have many uncertainty sources. But if you want to have the combined uncertainty, you don't add like $1 + 1$, because that would give you uncertainty of $2$, but that doesn't have to be the case. The true value of the measurement may lie in bet... |
Contents Introduction
In this chapter we will present and use a technique developed by the researchers Kay and Kajiya in 1986. Other acceleration structures since this time have proven to be better than their technique, but their solution can help to lay down the principles upon which most acceleration structures are b... |
Consider each of the following encryption schemes and state whether the scheme is perfectly secret or not. Justify your answer by giving a detailed proof if your answer is Yes, and a counterexample if your answer is No.
Consider an encryption scheme whose plaintext space is $\mathcal{M}=\{m\in\{0,1\}^\ell \mathrel{|} \... |
Is the following proof valid?
Let $X$ and $Y$ be jointly continuous random variables such that the joint density function is given by $$ f(x,y) = \begin{cases} ye^{-(x+y)} & \text{ for } x>0, y>0 \\ 0 & \text{ otherwise } \end{cases} $$
Then $X$ and $Y$ are dependent.
Proof
Let $X$ and $Y$ be continuous random variable... |
I'm reading the following set of notes on Taylor series and big O-notation, written by a professor at Columbia: http://www.math.columbia.edu/~nironi/taylor2.pdf. He repeatedly refers to what he calls "limit comparison", by which he means the theorem that for $a_n, b_n$ sequences of positive real numbers such that $b_n ... |
ISSN:
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eISSN:
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Discrete & Continuous Dynamical Systems - A
November 2002 , Volume 8 , Issue 4
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Abstract:
This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods... |
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Now showing items 1-3 of 3
D-meson nuclear modification factor and elliptic flow measurements in Pb–Pb collisions at $\sqrt {s_{NN}}$ = 5.02TeV with ALICE at the LHC
(Elsevier, 2017-11)
ALICE measured the nuclear modification factor ($R_{AA}$) and elliptic flow ($\nu_{2}$) of D mesons ($D^{0}$, $D^{+}$, $D^{⁎+}$... |
Cardinal numbers Cardinality is a measure of the size of a set. Two sets have the same cardinality---they are said to be equinumerous---when there is a one-to-one correspondence between their elements. The cardinality assignment problem is the problem of assigning to each equinumerosity class a cardinal number to repre... |
Is there any way to show that the following inequality holds for the given function with constraints?
$\frac{(a x + y)^{y+1}}{a x (a x + y + 1)^y}\geq 1$ for $0.5 \leq a \leq 1$, $x >0,y \geq 0$. It can be easily checked that it saturates the inequality for $\lim_{x \rightarrow \infty}$.
Numerically, I've checked this ... |
As mentioned above, a pseudorandom distribution “looks uniform” to all polynomial-time computations. We already know of a distribution that “looks uniform” — namely, the uniform distribution itself! A more interesting case is when \(\lambda\) uniform bits are used to
deterministically ( i.e., without further use of ran... |
Most of the permutation and combination problems we have seen count choices made without repetition, as when we asked how many rolls of three dice are there in which each die has a different value. The exception was the simplest problem, asking for the total number of outcomes when two or three dice are rolled, a simpl... |
, and to attain this field in specific regions of the brain, the electric current should pass through different head layers via skin, fat, skull, meninges, and cortex (part of the brain). In order to model the brain, different layers should be considered, including gray and white matters.The meninges, three layers of p... |
So not quite sure what you're missing, but here's how you go about doing this sort of thing.
So first, I am assuming this is 1D due to your description. Second, I'm assuming you know the relationship between points in the physical domain, $x$, and the computational domain, $\xi$, something along the lines of $x=x(\xi)$... |
Note that, by definition, the projections $P_n$ converge strongly to $I$.
Let $r\in\mathbb N$ (to be determined later), and define$$Q_n=P_{n+r}-P_n. $$The projections $Q_n$ are finite-rank, and pairwise orthogonal. Let $$S=\sum_n Q_nTQ_n,\ \ \ \ K=T-S.$$Let us check first that $SP_n=P_nS$ for all $n$. It is obvious tha... |
I agree with everything that's been said about Euler's formula not being a practical way of testing for primality, but it occurs to me there might be special numbers for which it
could be useful. Indulge me in a laborious "proof" that $n=82$ is not a prime.
Euler's formula in this case says
$$\begin{align}\sigma(82) = ... |
In section 4.5.1 of Nielsen and Chuang, two-level unitary matrices are defined as unitary matrices which act non-trivially only on two or fewer vector components. I'm not sure that I understand this definition. For instance, is $$\begin{pmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & 0 & \fr... |
Reinhardt
The existence of
Reinhardt cardinals has been refuted in $\text{ZFC}_2$ and $\text{GBC}$ by Kunen (Kunen inconsistency), the term is used in the $\text{ZF}_2$ context, although some mathematicians suspect that they are inconsistent even there. Definitions
A
weakly Reinhardt cardinal(1) is the critical point $... |
[latexpage]
NOTE: This article does not represent the whole truth. Actually OpenMP is better, not all relevant things have been exploited. There will be an updated article as soon as possible!
Many problems can be reduced, or reformulated as such that the solution is equivalent to solving a linear system of equations (... |
I have heard various talks at my institution from experimentalists (who all happened to be working on superconducting qubits) that the textbook idea of true "Projective" measurement is not what happens in real-life experiments. Each time I asked them to elaborate, and they say that "weak" measurements are what happen i... |
$$\textbf{I have been given the following:}$$
This is an example solution of a second order ODE, specifically that of a Harmonic Oscillator $$ {\text{d}v\over\text{d}t}=-\omega^2x-kv, $$ where $v=$d$x/$d$t$ and $\omega$ & $k$ are constants.
We can make this dimensionless by putting $x=x_0X$, $v=(x_0/t_0)V$ and $k=K/t_0... |
(similar to Mariano's post)
Q1: no. There are topological manifolds that don't admit triangulations, let alone smooth structures. All smooth manifolds admit triangulations, this is a theorem of Whitehead's. The lowest-dimensional examples of topological manifolds that don't admit triangulations are in dimension 4, the ... |
adjclust package
This document has two parts:
the first part aims at clarifying relations between dissimilarity and similarity methods for hierarchical agglomerative clustering (HAC) and at explaining implementation choices in
adjclust;
the second part describes the different types of dendrograms that are implemented i... |
Suppose I have a damped harmonic oscillator which is at rest, sitting comfortably with no initial amplitude, obeying the equation
$$\ddot{x} + \frac{1}{Q}\dot{x} + x = 0$$
where x is the vertical amplitude and Q is the quality factor. At $t = 0$, $x = 0$.
Now, suppose I model my system to include some sort of small per... |
Is there a "simple" mathematical proof that is fully understandable by a 1st year university student that impressed you because it is beautiful?
closed as primarily opinion-based by Daniel W. Farlow, Najib Idrissi, user91500, LutzL, Jonas Meyer Apr 7 '15 at 3:40
Many good questions generate some degree of opinion based... |
Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data
1.
Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
2.
Dipartimento di Matematica e Geoscienze Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy
In this ... |
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C=∫03π[1−cos2x121−sin2x12+1] dxC = \int_0^{3\pi} \bigg [ \frac{1 - \cos^2{\frac{x}{12}}} {1 - \sin^2{\frac{x}{12}}} + 1 \bigg ] \, \mathrm{d}xC=∫03π[1−sin212x1−cos212x+1]dxIf the value of 2+C1442 + \frac{C}{144}2+144C is in the form a/ba/ba/b where aaa and bbb ar... |
Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. Denote by $\mathcal P_n(X)$ the set of partition-valued functions on $X$ of total size $n$, i.e. the set of functions $\... |
The basis of the first derivative test is that if the derivative changes from positive to negative at a point at which the derivative is zero then there is a local maximum at the point, and similarly for a local minimum. If $f'$ changes from positive to negative it is decreasing; this means that the derivative of $f'$,... |
When we first considered what the derivative of a vector function might mean, there was really not much difficulty in understanding either how such a thing might be computed or what it might measure. In the case of functions of two variables, things are a bit harder to understand. If we think of a function of two varia... |
How to Model Fluid Friction in Joints with COMSOL Multiphysics®
Various machinery, such as engines, pumps, and turbines, employ components that transmit the load between the solid parts that are in relative motion. Common examples are piston rings, cams, gear teeth, and (of course) bearings. Often, these components are... |
Let $g \colon [0,\infty) \to \mathbb{R}$ be a monotonous function.
Suppose $g$ only attains positive values and is (not necessarily strictly) decreasing.
Does the sequence of series $$ s_k := \sum_{n=1}^\infty 2^{-k} g(n 2^{-k}) $$ converge to $$ \int_0^\infty g(t) dt $$ for $k \to \infty$?
Since $g$ is positive, the s... |
The identity$$E(\mathbf{r},t)=A(\mathbf{r},t)e^{i(\langle\omega\rangle t-\phi(\mathbf{r},t))}\tag{1}$$needs neither derivation nor justification; instead, it acts as an Ansatz for the electric field and as a definition for the pair of functions$$A(\mathbf{r},t)e^{-i\phi(\mathbf{r},t)}:=E(\mathbf{r},t)e^{-i\langle\omega... |
Fitting a logistic regression (
LR) model (with Age, Sex and Pclass as predictors) to the survival outcome in the Titanic data yields a summary such as this one:
##
## Call:
## glm(formula = Survived ~ Age + Sex + Pclass, family = binomial(link = logit),
## data = NoMissingAge)
##
## Deviance Residuals:
## Min 1Q Media... |
I know we can find out the electric field using the electric field $$E=\frac{KQ}{R^2}$$ taking small element $dq$ and finding the electric field by integrating the value of $dE$ over the circumference which will be $$E=\frac{kxQ}{\sqrt {(a^2+x^2)^3}}$$ where $a$ is radius of ring and $x$ is distance of point $p$ on the... |
253 23 Homework Statement There is an infinite charged plate in yz plane with surface charge density ##\sigma = 8*10^8 C/m^2## and negatively charged particle at coordinate (4,0,0) Find magnitude of efield at coordinate (4,4,0) Homework Equations E= E1+E2
So I figured to get e-field at point (4,4,0), I need to find the... |
历史查询
为了更好的帮助您理解掌握查询词或其译词在地道英语中的实际用法，我们为您准备了出自英文原文的大量英语例句，供您参考。
It is also shown that on the nilmanifold $\Gamma\backslash (H^3\times H^3)$ the balanced condition is not stable under small deformations.
For $\gamma\in\mathbb{R}$ let $C(\gamma)$ be the set of all $f\in{\mathcal S}^\prime$ for which$\sum_{n=0}^{\infty}\,|... |
I am reading an intro book about cryptography and the author tries to explain why using pseudo random number generators is vulnerable.
Given PRNG equation;
\begin{align} S_0 &= \text{seed}\\ S_{i+1} &\equiv A\cdot S_i + B \mod m, i = 0,1,\ldots \end{align}
where we choose $m$ to be 100 bits long and $S_i,A,B \in \{0,1,... |
One can make use of Simplify with AssumptionsI. Compute the sums=Sum[HarmonicNumber[n,5]/n^8,{n,1,Infinity}](* -(1/63) π^6 Zeta[7]-13/15 π^4 Zeta[9]-55 π^2 Zeta[11]+644 Zeta[13] *)II. Make a table of Zeta-functions with even argumentst=Flatten[Table[{ζ[2n]==Zeta[2n]},{n,0,6}]](* {ζ[0]==-(1/2),ζ[2]==π^2/6,ζ[4]==π^4/90,ζ... |
The equilibrium constant is known as \(K_{eq}\). A common example of \(K_{eq}\) is with the reaction:
\[aA + bB \rightleftharpoons cC + dD\]
\[K_{eq} = \dfrac{[C]^c[D]^d}{[A]^a[B]^b}\]
where:
At equilibrium, [A], [B], [C], and [D] are either the molar concentrations or partial pressures. Products are in the numerator. ... |
Q4.1
Write the Schrödinger equation for a particle in a two dimensional box with infinite potential barriers and adjacent sides of unequal length (a rectangle). Solve the equation by separating variables with a product function X(x)Y(y) to obtain the wavefunctions X(x) and Y(y) and energy eigenvalues. How many differen... |
I'd like to be able to auto-number equations and label them so i can refer to them in the text. Math Jax provides a built-in way to do this by setting the autoNumber option to AMS. I realize that by adding this option to the universal template, existing code will be affected in such a way that some equations will get u... |
If x(t) is even, then $x(t) = a_0 + \sum_{n=1}^{\infty}a_n*\cos(2\pi nt/T)$
However, based on this formula: $x(t) = a_0 + \sum_{n=1}^{\infty}a_n*\cos(2\pi nt/T) + b_n*\sin(2\pi nt/T)$ where $a_n = 2/T \int_o^Tx(t)*\cos(2\pi nt/T)$
x(t) is an even function and cos is an odd function. An even * odd = odd function. The pe... |
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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 ... |
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If the roots of 8x2−10x+3=08x^2-10x+3=08x2−10x+3=0 are α\alphaα and β2\beta^2β2 where β2>12\beta^2>\frac{1}{2}β2>21, then find the equation whose roots are (α+iβ)100(\alpha+i\beta)^{100}(α+iβ)100 and (α−iβ)100(\alpha-i\beta)^{100}(α−iβ)100.
Problem Loading...
Not... |
Theorem
The following definitions of the concept of
Compact Space in the context of Topology are equivalent:
A topological space $T = \left({S, \tau}\right)$ is
compact if and only if every open cover for $S$ has a finite subcover.
A topological space $T = \left({S, \tau}\right)$ is
compact if and only if $\tau$ has a ... |
Answer
The solution set is $\{\varnothing\}$
Work Step by Step
$$\tan^2x+3=0$$ over interval $[0,2\pi)$ 1) Consider the equation: $$\tan^2x+3=0$$ $$\tan^2x=-3$$ We know that $A^2\ge0$ for $\forall A\in R$. As a result, $\tan^2x\ge0$ for $\forall x\in [0,2\pi)$ Therefore, as $-3\lt0$, there are no values of $x\in[0,2\pi... |
Relative Complement inverts Subsets Theorem
Let $S$ be a set.
Let $A \subseteq S, B \subseteq S$ be subsets of $S$.
Then: $A \subseteq B \iff \relcomp S B \subseteq \relcomp S A$
where $\complement_S$ denotes the complement relative to $S$.
Proof
\(\displaystyle A\) \(\subseteq\) \(\displaystyle B\) \(\displaystyle \le... |
Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity
1.
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730
2.
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706
(*)$ \qquad\qquad \varphi(x)\ge\varphi_{\star \star}(x) \eq... |
I confused with a question from the past-paper of Lagrangian and Hamiltonian mechanics. A Lagrangian (plane polar coordinate) for the spaceship (mass is $m$) under influence of central force directed towards the centre of Earth is $$L=\frac{1}{2} m \left( \dot{r}^2 +r^2 \dot{\phi}^2\right)+\frac{k}{r}$$
Where the $k$ i... |
Set is Subset of Itself
Jump to navigation Jump to search
Theorem $\forall S: S \subseteq S$ Proof
\(\displaystyle \forall x: \ \ \) \(\displaystyle (x \in S\) \(\implies\) \(\displaystyle x \in S)\) Law of Identity: a statement implies itself \(\displaystyle \leadsto \ \ \) \(\displaystyle S\) \(\subseteq\) \(\display... |
V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem.” (manuscript under review)
@ARTICLE{GitmanHamkinsKaragila:KM-set-theory-does-not-prove-the-class-Fodor-theorem, author = {Victoria Gitman and Joel David Hamkins and Asaf Karagila}, title = {Kelley-Morse set theor... |
J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
@ARTICLE{HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD, author = {Joel David Hamkins and Jonas Reitz}, title = {The se... |
Recap of Lecture 18
Last Lecture addressed the angular moment of an electron revolving around the nucleus. This is described by the \(l\) quantum number and an electron in any non-spherical orbital (i.e., an s orbital) will have an angular moment (you should know formula). The \(m_l\) quantum number designates the orie... |
Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.
Are there good examples of \begin{equation} \lim_{x \to c} f(x) \neq f(c), \end{equation} or of cases when $c$ is not in the domain of $f(x)... |
Here is an answer to the extra question regarding regular rings:
A finitely generated module is flat if and only if it is locally free, so for finitely generated modules your question translates to:
Q1 When is a torsion-free module/sheaf locally free?
and
Q2 What is a simple example of a torsion-free (say coherent) she... |
Comparing Two Interfaces for High-Frequency Modeling
It is always important to choose the correct tool for the job, and choosing the correct interface for high-frequency electromagnetic simulations is no different. In this blog post, we take a simple example of a plane wave incident upon a dielectric slab in air and so... |
I'm a bit stumped on this one.
Show that $\lim_{x\to0} \frac{e^x -1}{\sin(x)} = 1$ using power series.
The instructions are not to use L'Hospital's Rule. I cannot find a way to do this without L'Hopital even simplifying using series expansion.
Mathematics Stack Exchange is a question and answer site for people studying... |
J. D. Hamkins, “Every countable model of set theory embeds into its own constructible universe,” J. Math. Logic, vol. 13, iss. 2, p. 1350006, 27, 2013.
@article {Hamkins2013:EveryCountableModelOfSetTheoryEmbedsIntoItsOwnL,
AUTHOR = {Hamkins, Joel David},
TITLE = {Every countable model of set theory embeds into its own
... |
Slingshot argument From The Art and Popular Culture Encyclopedia
Featured:
This type of argument was dubbed the "slingshot" by philosophers Jon Barwise and John Perry (1981) due to its disarming simplicity. It is usually said that versions of the slingshot argument have been given by Gottlob Frege, Alonzo Church, W. V.... |
This is a little question inspired from Hartshorne's Geometry, which I've been juggling around for a while.
Suppose that $\Pi$ is the Cartesian plane $F^2$ for some field $F$, with the set of ordered pairs of elements of $F$ being the points and lines those subsets defined by linear equations. Let $\Pi'$ be the associa... |
A
local maximum point on a function is apoint $(x,y)$ on the graph of the function whose $y$ coordinate islarger than all other $y$ coordinates on the graph at points "closeto'' $(x,y)$. More precisely, $(x,f(x))$ is a local maximum if thereis an interval $(a,b)$ with $a< x< b$ and $f(x)\ge f(z)$ for every $z$in $(a,b)... |
I understand an MDP (Markov Decision Process) model is a tuple of $\{S, A, P, R \}$ where:
$S$ is a discrete set of states $A$ is a discrete set of actions $P$ is the transition matrix ie. $P(s' \mid s, a) \rightarrow [0,1]$ $R$ is the reward function id. $R(s, a, s') \rightarrow \mathbb{R}$
For a non-trivial MDP, say ... |
Although the conversion of one element to another is the basis of natural radioactive decay, it is also possible to convert one element to another artificially. The conversion of one element to another is the process of transmutation. Between 1921 and 1924, Patrick Blackett conducted experiments in which he converted a... |
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Description
We give an explicit construction of a pseudorandom generator for read-once formulas whose inputs can be read in arbitrary order. For formulas in $n$ inputs and arbitrary gates of fan-in at most $d = O(n... |
I want to determine whether the following statement is true:
Let $R$ be any ring, and $V$ an $R$-module that is a union $V=\bigcup_{n=1}^\infty V_n$ of submodules $V_1 \subseteq V_2 \subseteq \dots$. If each $V_i$ is projective, then so is $V$.
My attempt at a proof is to use Zorn's lemma as follows: let $g : M \twohea... |
This is simply proportional to the 3-tachyon correlation function. They warn you above the equation that the momentum conservation factor such as $$(2\pi)^D\delta^{(D)}(k_1-k_2-k_3)$$ is omitted everywhere and I am not even sure whether their normalization of the states includes the power of $2\pi$ factor. Pick Polchin... |
Suppose n is a positive integer such that 6n has exactly 9 positive divisors.
How many prime numbers are divisors of 6n?
\(6=2^13^1\\ \text{let }n = \prod \limits_{k=0}^\infty p_k^{n_k} \\ \text{where }p_k \text{ is the }kth \text{ prime}\\ 6n = 2^{n_2+1}\cdot 3^{n_3+1}\cdot \prod \limits_{k=3}^\infty p_k^{n_k} \)
\(6n... |
What does it mean when the price elasticity of demand %Qd/%P is greater than one? Typically I hear that it means the demand is elastic since if, say, the price decreases by 1% the demand for the good increases by more than 1%. But what happens if %P is +1% and %Qd still increases by more? Sure, this is elastic, but doe... |
It is common in physics literature to identify invariant geometric objects with their components in some reference frame.
For example, if $T$ is a type (2,0) tensor field, then it is common to refer to its components $T^{\mu\nu}$ as a tensor field.
In essence, the connection coefficients are the components of a connect... |
Gamma is the second partial derivative of the change in the price of the option wrt to the change in the underlying. Said another way, it is the change in delta. If you write down the Black-Scholes pricing formula, you's see the gamma term:$$...\frac{1}{2}\frac{\partial^2C}{\partial S^2}(\Delta S)^2...$$Notice that the... |
Let $V$ a vector space of finite dimension an let $T:V\rightarrow V$ be a linear operator such that every hyperplane of $V$ is $T$-stable. Prove $T=\lambda\,\mathit{Id}_{V}$ for some $\lambda$.
Note: $\mathit{Id}_{V}$ is the identity operator.
Proof:
Pick a vector $v \in V \setminus \{ 0 \}$. You can find $v_2, \dots, ... |
Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies:
$L(1) = 0$
$L(x) = 1$
$L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$
Is $L$ necessarily the derivative? Maybe if I throw in some kind of continuity assumption on $L$? If it helps you can throw the "chain rule" into the list of pr... |
Problem evaluating a very tiny integral
Hi everyone,
I'm pretty new with sage. I was forced to change from MATLAB to Sage, because I was told Sage does approximate very tiny numbers better as it can work with sqrt(2) as sqrt(2) and not as the rational number approximating it.
Approximations are very important for my pr... |
Fatima has done some previous analyses and has found that the stock price over any period of time can be modelled reasonably accurately with the following equation:\[ \operatorname {price}(k) = p \cdot (\sin (a \cdot k+b) + \cos (c \cdot k+d) + 2) \]
where $p$, $a$, $b$, $c$ and $d$ are constants. Fatima would like you... |
Consider a dielectric slab waveguide (lossless, isotropic) illuminated transversally from the vacuum (with coherent, monochromatic light).
We define the
base bandwidth of a waveguide (or optical fiber), $AB$, to be the inverse of the time retardation, $\Delta t$, at 1 km of the waveguide between the energy of a guided ... |
As we have defined it in Section 1.3, a function is a very general object. At this point, it is useful to introduce a collection of adjectives to describe certain kinds of functions; these adjectives name useful properties that functions may have. Consider the graphs of the functions in Figure 2.5.1. It would clearly b... |
As we begin to compile a list of convergent and divergent series, new ones can sometimes be analyzed by comparing them to ones that we already understand.
Example 13.5.1 Does $\ds\sum_{n=2}^\infty {1\over n^2\ln n}$ converge?
The obvious first approach, based on what we know, is the integral test. Unfortunately, we can... |
I am working on a physics reserach project for school and I have run into some troubles working Mathematica. I am a fairly inexperienced mathematica user so any help would very much be appreciated.
I need to find the roots of the transcendental equation $$\zeta_n \tan(\zeta) - \sqrt{R^2-\zeta^2_n}=0$$ and then collect ... |
There are many problems that a data scientist encounters when “fighting” financial data for the first time: nothing is normally distributed, most problems are tough (low signal to noise ratio) and non-stationary high-dimensional time series are ubiquitous.
In Quantdare we have spoken many times about one of the main so... |
The conceptually simplest way to produce a W state is somewhat analogous to classical reservoir sampling, in that it involves a series of local operations that ultimately create a uniform effect.
Basically, you look at each qubit in turn and consider "how much amplitude do I have left in the all-0s state, and how much ... |
I'm studying
Stochastic Processes by Richard F. Bass. Within this book I encountered the definition of a Markov process, which is given as follows:
We are given a separable metric space $S$ endowed with its Borel $\sigma$-field and a measurable space $(\Omega, \mathcal{F})$ together with a filtration $\{\mathcal{F}_t\}... |
I was observing here and conjecture $(1)$
$$\lim_{n\to\infty}{S_{n-1}S_{n+2}\over S_nS_{n+1}}=e^2\tag1$$
Given that Harlan Brothers' formula $(2)$
$$\lim_{n\to\infty}{S_{n-1}S_{n+1}\over S_n^2}=e\tag2$$
Trying to prove $(1)$:
$(1)\div(2)$
$$\lim_{n\to\infty}{S_{n-1}S_{n+2}\over S_nS_{n+1}}\times{S_n^2\over S_{n-1}S_{n+... |
Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$.
Question:Let $n$ be a positive integer which is not a perfect square. Is there always a polynomial $D \in \mathbb{Z}[x]$ of degree $2$, an integer $k$ a... |
Problem
Take this (easy) problem as an example:
An astronomer is interested in measuring the distance, in light-years, from his observatory to adistant star. Although the astronomer has a measuring technique, he knows that, because ofchanging atmospheric conditions and normal error, each time a measurement is made it w... |
Contents MA 453 Fall 2008 Professor Walther News
Here are some basic pointers:
In order to do any editing, you must be logged in with your Purdue career account. If you look under MediaWiki FAQ, you get lots of instructions on how to work with Rhea. Some important things are under item 4 in that manual. If you want to ... |
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