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Note that the question only makes sense when $T\geq T_{\rm c}$, since the 2-point function does not decay when $T<T_{\rm c}$ (neither in the half-plane, nor in the full plane).
I don't know how to prove the result when $T=T_{\rm c}$, although it is quite plausible that the technology developed around SLE makes it possi... |
Quite a lot of algebraic number theory was invented through trying to prove Fermat's last theorem and other Diophantine problems.
For example, if I asked you to solve the equation $x^2 - y^2 = 5$ in integers it is very simple, you can factorise $(x+y)(x-y) = 5$ and solve the problem by linking to divisors of $5$, in or... |
This is related to waves & optics.
I am given the fourier transform of a function (the spectrum of frequences for a pulse of sound)
$$\hat f(w) = sinc((w-w_0)\tau ) $$
And now, a filter is attached so that it only allows waves with frequency $w_0$ to passthrough, and am asked to find what f(t) is like after we put the ... |
Topologies on Sets
Definition: Let $X$ be a set. A Topology on $X$ is a collection $\tau$ of subsets of $X$ that satisfies the following properties: 1) $X, \emptyset \in \tau$. 2) If $\{ U_i : i \in I \}$ is any arbitrary collection of subsets of $X$ such that $U_i \in \tau$ for all $i \in I$ then the union $\displayst... |
Refine Year of publication 1998 (21) (remove) Document Type Article (21) (remove) Keywords
The Wannier-Bloch resonance states are metastable states of a quantum particle in a space-periodic potential plus a homogeneous field. Here we analyze the states of quantum particle in space- and time-periodic potential. In this ... |
This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential.
Whilst reading notes relating to Girsanov it is stated that the change of measure density martingale may be written: \begin{align} \rho_t = \exp \left[- \int_{0}^{t} \lambda_s \, dW_s - \tf... |
Focus Questions
The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts motivated by these questions and be able to write precise, coherent answers to these questions. What is de Moivre’s Theorem and why is it useful? If \(n\) ... |
The Closure of a Set Equals the Union of the Set and its Accumulation Points
Recall from The Closure of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the closure of $A$ is the smallest closed subset containing $A$ denoted $\bar{A}$.
We will now look at a very nice... |
The Dimension of a Sum of Subspaces Examples 1
Recall from The Dimension of a Sum of Subspaces page that if $V$ is a finite-dimensional vector space and if $U_1$ and $U_2$ are subspaces of $V$ then:(1)
We will now look at some example problems regarding this important formula for finite-dimensional vector spaces.
Examp... |
Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities
SISSA -International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy
$ u'' + \sum\limits_{i = 1}^m {} {\alpha _i}{a_i}(x){g_i}(u) - \sum\limits_{j = 1}^{m + 1} {} {\beta _j}{b_j}(x){k_j}(u) = 0,{\r... |
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. To see ten new questions, reload the page.
I used the official objectives and sample test to construct these questions, but cannot... |
Abstract: Bordered Floer homology is an invariant for three-manifolds with boundary, defined in collaboration with Robert Lipshitz and Dylan Thurston. The invariant associates a DG algebra to a parameterized surface, and a module over that algebra to a three-manifold with boundary. I will explain how methods from borde... |
655 results for "part".
Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots.
Question Needs to be tested
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Last modified 10/10/2019 13:43
No subjects selected No topics selected
No ability levels selected
The student is ask... |
I'm working on a project, and I have to use the cumulative and conditional expected value of the variations of a stock following a Geometric Brownian Motion.
I know that the cumulative is as follows : $$ \mathbb{E}\left[ \mathbb{1}_{ \frac{S_{i+1}}{S_{i}} < z}\right] = \mathbb{P} \left[ \frac{S_{i+1}}{S_{i}} < z \right... |
J. D. Hamkins, “book review of G.~Tourlakis, Lectures in Logic and Set Theory, vols.~I & II,” Bulletin of Symbolic Logic, vol. 11, iss. 2, p. 241, 2005.
@ARTICLE{Hamkins2005:TourlakisBookReview, AUTHOR = "Joel David Hamkins", TITLE = "book review of {G.~Tourlakis}, {Lectures in Logic and Set Theory}, vols.~{I \& II}", ... |
Antonella Altamura and Marco Bee spotted that the language of the discussion on tail index for ARCH type data was not correct. It said that \begin{equation*} \Gamma(\iota/2+1/2)=\sqrt{\pi}(2\alpha)^{-\iota/2} \end{equation*} was the unconditional distribution of which of course does not make sense.
Instead it should sa... |
No. Not at all.
If a language has Godel numbering (and certainly the language of set theory with only $\in$ has that), then asserting that a theory in that language is consistent is a number theoretic statement. Namely, it's a statement about integers.
Of course, we need to assume that the numbers encoding the axioms o... |
Metric Spaces Are Compact Spaces If and Only If They're Countably Compact
Recall from The Lebesgue Number Lemma page that if $(X, d)$ is a metric space that is also a BW space then for every open cover $\mathcal F$ of $X$ there exists an $\epsilon > 0$ called a Lebesgue number such that for all $x \in X$ there exists a... |
The word ‘ Trigonometry ’ is derived from the Greek word and the subject is developed to solve geometric problems involving triangles. It is used to measure the sides of a triangle. An angle is a measure of rotation of a given ray about its initial point and the original ray is called the initial side and the final pos... |
Let $\varphi: R \rightarrow S$ be a (unital) ring homomorphism. So every left $S$-module $M$ has also a left $R$-module structure via $\varphi$ and in general we have $$ \text{End}_S(M) \subseteq \text{End}_R(M)$$ My question is: Is there a necessary and sufficient condition on $\varphi$ such that the above inclusion b... |
Let $f$ be a non-negative Riemann integrable function on $[a,b]$. If $f$ equals to zero except on an null set,then$\int_a^b f = 0$
Let $A$ be the null set and $M=\sup\left\{f(x):x\in[a,b]\right\}$.
For any $\epsilon>0$, there exists a sequence of intervals $(I_k)$ such that $A\subset \bigcup_{k=1}^\infty I_k$ and $\sum... |
I am reading about testing independence in two-way contingency tables from Mood Graybill and Boes's Introduction to the Theory of Statistics and is confused about testing independence.
We have a two-way contingency table. We assume that the cells in the table follow a multinomial distribution with parameters $n$ (known... |
Here we’ll look at some more probability calculations before moving on to permutations, combinations and the binomial theorem. More Probability…
If you recall from the last part, we used set notation to describe a general way of calculating simple probability: \(P(A)={{|A|} \over {|S|}}.\) Where \(A\) is a set of
event... |
+
Recall from The nth Convergent of an Infinite Continued Fraction page that if $\langle a_0; a_1, a_2, ... \rangle$ is an infinite simple continued fraction with $a_0 \in \mathbb{Z}$ and $a_n \in \mathbb{N}$ for $n \geq 1$ then the $n^{\mathrm{th}}$ convergent of this infinite simple continued fraction is defined to b... |
I'm not sure about your calculation but Matlab yields the correct result. In this problem, the common approach is to use
Routh-Hurwitz criterion and search for a row of zeros that yields the possibility for imaginary axis roots. For convert the system to the closed-loop transfer function, hence $$\frac{K}{s^4 + 10s^3 +... |
I am using TTR in R and I am trying to understand the Yang Zhang volatility estimator (without drift). The following equations seem to imply a single value:
$$ \sigma = \sqrt{{\sigma_o^2}+k\sigma_c^2+(1-k)\sigma_{rs}^2} $$
$$\sigma_o^2 = \frac{1}{N-1}\sum_{i=1}^{N}ln{\frac{o_i}{c_{i-1}}^2}$$ $$\sigma_c^2 = \frac{1}{N-1... |
I asked the following question in MSE for which I couldn't get any answer yet. I thought this would be a better place for that question.
In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\in... |
Section 6.1 Exercises
1. Sketch a graph of \(f\left(x\right)=-3\sin \left(x\right)\).
2. Sketch a graph of \(f\left(x\right)=4\sin \left(x\right)\).
3. Sketch a graph of \(f\left(x\right)=2\cos \left(x\right)\).
4. Sketch a graph of \(f\left(x\right)=-4\cos \left(x\right)\).
For the graphs below, determine the amplitud... |
Periodic attractors of nonautonomous flat-topped tent systems
ISEL - Instituto Superior de Engenharia de Lisboa, Mathematics Department and CIMA - Research Centre for Mathematics and Applications, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal
In this work we will consider a family of nonautonomous dynami... |
In Boyd's Convex Optimization, pp. 243,
for
anyoptimization problem ... for which strong duality obtains, any pair of primal and dual optimal points must satisfy the KKT conditions
i.e. $\mathrm{strong ~ duality} \implies \mathrm{KKT ~ is ~ necessary ~ condition ~ for ~ optimal ~ solution}$
and in pp. 244,
(When the pr... |
I am learning set theory and I am curious if we could have unrestricted comprehension while blocking Russell's paradox using the axiom of regularity/foundation. To my very limited knowledge, axiom of regularity is not needed to block paradoxes since if ZF without regularity is inconsistent, then adding regularity would... |
Table of Contents
Locally Connected and Locally Path Connected Topological Spaces
Recall from the Connected and Disconnected Topological Spaces page that a topological space $X$ is said to be connected if it is not disconnected.
Also recall from the Path Connected Topological Spaces page that a topological space $X$ is... |
187 results for "into".
Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.
Question Ready to use
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What is the value of the expression g... |
Differential Equations also called as Partial differential equations if they have partial derivatives. The highest order derivative is the order of differential equation.
Differential Equation formula
\(\frac{dy}{dt} + p(t)y = g(t)\)
p(t) & g(t) are the functions which are continuous.
y(t) = \(\frac{\int \mu (t)g(t)dt ... |
Learning Objectives
In this section, you will:
Use the Law of Cosines to solve oblique triangles. Solve applied problems using the Law of Cosines. Use Heron’s formula to find the area of a triangle.
Suppose a boat leaves port, travels \(10\) miles, turns \(20\) degrees, and travels another 8 miles as shown in Figure \(\... |
Equipartition of energy for nonautonomous wave equations
1.
Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 337, Memphis, TN 38152, USA
2.
Department of Mathematical Sciences, The University of Memphis, Dunn Hall, 343, Memphis, TN 38152, USA
3.
Department of Mathematics, Statistics and Physic... |
Let $G$ be a finite solvable group and $N$ be a normal subgroup of $G$ such that $G/N$ is not abelian. Also for every prime integer $p$, $G$ has at most $5$ conjugacy classes whose sizes are multiples of $p$. Moreover, let $G/N$ be a Frobenius group with kernel $K/N$ of order $5$ and complement isomorphic to $G/K$ of o... |
May 2nd, 2015, 05:23 AM
# 1
Newbie
Joined: May 2015
From: Imperium Romanum
Posts: 13
Thanks: 0
Need help with Lagrange multiplier...
Hi Everyone,
How do I workout the partial derivative for 1.) and the workings to the solution 2.)?
Your help is much appreciated!
May 2nd, 2015, 06:21 AM
# 2
Math Team
Joined: Jan 2015
Fr... |
I could need some advice on extensions of the CIR model.
The standard CIR reads
$dr(t)=\kappa(\theta-r(t))dt + \sigma \sqrt{r(t)} dW(t)$.
A possible extension, if we would like the short-rate to also include negative values, could be a displaced version, so that $r(t)+\alpha$, where $\alpha>0$, follows a CIR model.
Fur... |
Binomial Tree Simulation
The binomial model is a discrete grid generation method from \(t=0\) to \(T\). At each point in time (\(t+\Delta t\)) we can move up with probability \(p\) and down with probability \((1-p)\). As the probability of an up and down movement remain constant throughout the generation process, we en... |
Question 1:
The Boltzmann entropy $S_B=k_B\ln\Omega(E)$ is valid only for the microcanonical ensemble. In the microcanonical ensemble, all accessible microstates (accessible = they have energy $E$, at least with some $\delta E$ uncertainty) have equal probability. So if $r$ is an index that labels microstates, we have ... |
$\vec{B}=\nabla \times \vec{A}\tag1$
This is true because at
every point $\nabla\cdot\vec{B}=0 \tag2$
In
free space points,
$\displaystyle \vec{B}=\dfrac{\mu_0}{4 \pi}\int_C \dfrac{I\ dl \times\hat{r}}{r^2}\tag3$
Consequently: $\nabla \cdot\vec{B}=0 \tag2$
At the
points on the circuit, there is a singularity and we can... |
PLaces text as a title, xlabel, or ylabel on a group of subplots. Returns a handle to the label and a handle to the axis.
[ax,h]=suplabel(text,whichLabel,supAxes)
returns handles to both the axis and the label.
ax=suplabel(text,whichLabel,supAxes)
returns a handle to the axis only. suplabel(text) with one input argumen... |
Contact InfoPure Mathematics
University of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x33484 Fax: 519 725 0160 Email: puremath@uwaterloo.ca Alessandro Portaluri, University of Turin
"Existence and Stability Results in Celestial Mechanics"
Is th... |
To do it for a particular number of variables is very easy to follow. Consider what you do when you integrate a function of x and y over some region. Basically, you chop up the region into boxes of area ${\rm d}x{~\rm d} y$, evaluate the function at a point in each box, multiply it by the area of the box. This can be n... |
Difference between revisions of "Unitriangular matrix group:UT(3,p)"
(→External links)
(→As a group of matrices)
Line 5: Line 5:
===As a group of matrices===
===As a group of matrices===
−
Given a prime <math>p</math>, the group <math>UT(3,p)</math> is defined as the [[unitriangular matrix group]] of [[unitriangular ma... |
If I understand correctly, a distribution in the exponential family... $$\underline X\sim f_{\underline\theta}(\underline x) = exp\{\sum\limits_{i}\eta_i(\underline\theta)T_i(\underline x)-B(\underline\theta)\}~h(\underline x)$$ ...where $\underline\eta(\centerdot)$ is a (possibly vector valued) parameter, $\underline ... |
In set theory, we have the phenomenon of the
universal definition. This is a property $\phi(x)$, first-order expressible in the language of set theory, that necessarily holds of exactly one set, but which can in principle define any particular desired set that you like, if one should simply interpret the definition in ... |
And I think people said that reading first chapter of Do Carmo mostly fixed the problems in that regard. The only person I asked about the second pset said that his main difficulty was in solving the ODEs
Yeah here there's the double whammy in grad school that every grad student has to take the full year of algebra/ana... |
The transistor going into saturation isn't a property of the transistor itself, but instead a property of the circuit surrounding the transistor
the transistor, as part of it. and
The simplest case to imagine is an NPN switch. I'll present two different such switch circuits to make the above point concretely clear:
sim... |
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Now showing items 1-2 of 2
Anisotropic flow of inclusive and identified particles in Pb–Pb collisions at $\sqrt{{s}_{NN}}=$ 5.02 TeV with ALICE
(Elsevier, 2017-11)
Anisotropic flow measurements constrain the shear $(\eta/s)$ and bulk ($\zeta/s$) viscosity of the quark-gluon plasma created in heavy-ion collisions... |
It should come as no surprise that we can use this reasoning about division in the “Dots & Boxes” model in other bases as well.
The following picture shows that working in base 5,
$$1432_{five} \div 13_{five} = 110_{five} R2_{five},\; \text{meaning}\; 1432_{five} = 110_{five} \cdot 13_{five} + 2_{five} \ldotp$$
Think /... |
One general rule about technical papers--especially those found on the Web--is that
the reliability of any statistical or mathematical definition offered in them varies inversely with the number of unrelated non-statistical subjects mentioned in the paper's title. The page title in the first reference offered (in a com... |
diff options
Diffstat (limited to 'docs')
-rw-r--r-- docs/tutorial/solver.md 79
1 files changed, 78 insertions, 1 deletions
diff --git a/docs/tutorial/solver.md b/docs/tutorial/solver.md
index 17f793e..b150f64 100644
--- a/docs/tutorial/solver.md
+++ b/docs/tutorial/solver.md
@@ -6,7 +6,14 @@ title: Solver / Model Opti... |
In Exercises \((2.2E.1)\) to \((2.2E.12)\), find the general solution.
Exercise \(\PageIndex{1}\)
\(y''+5y'-6y=0\)
Answer
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{2}\)
\(y''-4y'+5y=0\)
Answer
Add texts here. Do not delete this text first.
Exercise \(\PageIndex{3}\)
\(y''+8y'+7y=0\)
Answer
Ad... |
Since the limit $\frac{\sin(x)}{x}=1$ for $x \rightarrow 0$, I wondered about the infinite product:
$$\prod^{\infty}_{n=1} n \sin \left( \frac{1}{n} \right)=\sin(1) \cdot 2 \sin\left( \frac{1}{2} \right) \cdot 3 \sin\left( \frac{1}{3} \right) \dots$$
By numerical experiment in Mathematica it seems to converge, even if ... |
base
Geometric construction of the global base of the quantum modified algebra of
However the compatibility of the canonical base of the modified algebra and of the geometric base given by intersection cohomology sheaves on the affine flag variety was never proved.
We prove that these determinantal semi-invariants span... |
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... |
In a paper by Joos and Zeh, Z Phys B 59 (1985) 223, they say:This 'coming into being of classical properties' appears related to what Heisenberg may have meant by his famous remark [7]: 'Die "Bahn" entsteht erst dadurch, dass wir sie beobachten.'Google Translate says this means something ...
@EmilioPisanty Tough call. ... |
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Now showing items 1-10 of 24
Production of Σ(1385)± and Ξ(1530)0 in proton–proton collisions at √s = 7 TeV
(Springer, 2015-01-10)
The production of the strange and double-strange baryon resonances ((1385)±, Ξ(1530)0) has been measured at mid-rapidity (|y|< 0.5) in proton–proton collisions at √s = 7 TeV with the ... |
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Now showing items 1-5 of 5
Measurement of electrons from beauty hadron decays in pp collisions at root √s=7 TeV
(Elsevier, 2013-04-10)
The production cross section of electrons from semileptonic decays of beauty hadrons was measured at mid-rapidity (|y| < 0.8) in the transverse momentum range 1 < pT <8 GeV/c wit... |
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Now showing items 11-20 of 27
Pseudorapidity dependence of the anisotropic flow of charged particles in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
(Elsevier, 2016-11)
We present measurements of the elliptic ($\mathrm{v}_2$), triangular ($\mathrm{v}_3$) and quadrangular ($\mathrm{v}_4$) anisotropic azimutha... |
I’m working on a number theory proof that has been giving me some trouble for a while. I will explain the problem and the attempts I’ve made.
Let $x\in \mathbb{R}$ and $d \in \mathbb{Z}$ where both $x, d > 0$ (i.e. positive values). Prove that the number of integers, say k, that are $\leq $ $x$ and divisible by $d$ is ... |
Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases
1.
Institute of Communications and Navigation, German Aerospace Center (DLR), D-82234 Oberpfaffenhofen, Germany
2.
Institute for Communications Engineering, Technical University of Munich (TUM), D-8029... |
Here is a proof in the real case. For general fields, this only gives a lower bound of $\lceil n/2 \rceil$, though the correct lower bound should indeed be $n$. For more, take a look at the monograph
Algebraic complexity theory by Bürgisser, Clausen and Shokrollahi. Model
It will be easier to give a lower bound on comp... |
Wikipedia offers the following definition for an (embedded) submanifold:
An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a topological embedding.
I've been wondering if one could not equivalently define a submanifold like this:
$(\ast)$ Let $M$ be a... |
Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity :
$${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$
with $$v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}$$ and
$$V^{\nu}=\dfrac{\text{d}x^{\nu}}{\t... |
Jan 13, 2019
Great course for kickoff into the world of CNN's. Gives a nice overview of existing architectures and certain applications of CNN's as well as giving some solid background in how they work internally.
Sep 02, 2019
This is very intensive and wonderful course on CNN. No other course in the MOOC world can be ... |
The second formula is wrong: the outside parts are equal to each other, but the middle part is merely proportional to (and not necessarily equal to) the outside parts. The likelihood is defined by $L(\theta \mid y) = k(y) p(y \mid \theta) \propto p(y \mid \theta)$ where $k$ is some constant-of-proportionality that does... |
Journal of Symbolic Logic J. Symbolic Logic Volume 65, Issue 3 (2000), 1223-1240. Fragments of Heyting Arithmetic Abstract
We define classes $\Phi_n$ of formulae of first-order arithmetic with the following properties: (i) Every $\varphi \in \Phi_n$ is classically equivalent to a $\Pi_n$-formula (n $\neq$ 1, $\Phi_1 :=... |
The Kunen inconsistency The Kunen inconsistency, the theorem showing that there can be no nontrivial elementary embedding from the universe to itself, remains a focal point of large cardinal set theory, marking a hard upper bound at the summit of the main ascent of the large cardinal hierarchy, the first outright refut... |
The Open and Closed Sets of a Topological Space
Consider a topological space $(X, \tau)$. We will now define exactly what the open and closet sets of this topological space are.
Definition: Let $(X, \tau)$ be a topological space. If $A \subseteq X$ is such that $A \in \tau$ then $A$ is said to be Open. A subset $A \sub... |
Complex Roots of The Characteristic Equation
Consider the following second order linear homogenous differential equation $a \frac{d^2 y}{dt^2} + b \frac{dy}{dt} + cy = 0$ where $a$, $b$, and $c$ are constants. Recall that the characteristic equation for this differential equation is the quadratic polynomial $ar^2 + br ... |
Let $X$ be a Hausdorff locally compact in $x \in X$. Show that for each open nbd $U$ of $x$ there exists an open nbd $V$ of $x$ such that $\overline{V}$ is compact and $\overline{V} \subset U$.
My work:
Since $X$ is Hausdorff and locally compact then $X$ is regular. Let $U$ be an open nbd of $x$. By assumption $X$ is l... |
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... |
If $A \sim B$ and $B \sim C$. Prove that $A \sim C$
What I have:
We know there exists functions $f,g$ such that $f:A\to B$ and $g:B \to C$ where $f$ and $g$ are bijective.
We thus require to show that there exists a function $h:A \to C$ where $h$ is bijective.
Can anyone please give me some hints that might point me in... |
In addition to the excelent answers already given, there are a few subleties one should explicitly point out.
There are two main concepts for integration, the first being
indefinite integration, that is finding the antiderrivative of a function, and the second is definite integration, finding the measure of the (signed... |
a) Regular implies normal.
Okay, maybe that's too high-tech. If $f=g+hy$ lies in the integral closure, then it satisfies the quadratic polynomial $(z-g)^2 - h^2(x^3-x)$. If the coefficients are polynomials (which they must be, by Gauss's Lemma), then $g$ is a polynomial, and $h^2 (x^3-x)$ is a polynomial. But $x^3-x$ i... |
Suppose $i$ and $j$ are indices which take values $1, \dots, m$, and for each $i$ and $j$, we have a number $a_{ij}$. Note that $(i, j) \in \{1, \dots, m\}\times\{1, \dots, m\}$. If we were to sum over all possible values of $(i, j)$ we would have
$$\sum_{(i, j) \in \{1, \dots, m\}\times\{1, \dots, m\}}a_{ij}$$
which c... |
Perhaps you should revisit the definitions of random variable and distribution to clarify things. For random variables, I like the one on Wikipedia for its simplicity.
A random variable $X : \Omega \rightarrow E$ is a measurable function from a set of possible outcomes $\Omega$ to a measurable space $E$.
On the other h... |
A field $\phi(z)$ has the conformal weight $h$, if it transforms under $z\rightarrow z_1(z)$ as
$$ \phi(z) = \tilde{\phi}(z_1)\left(\frac{dz_1}{dz}\right)^h $$
The (classical) scaling dimension can be obtained for each field by appearing in the Lagrangian by making use of the constraint that has to be dimensionless, re... |
Sources of Error
Sources of Error
It is always important to acknowledge possibly sources of error - especially when it comes to applied mathematics dealing with biology, physics, chemistry, engineering, economics, etc… We will now outline some of the sources of error.
Calculation Errors. This is the most obvious type o... |
Let $m^*$ denote the outer measure corresponding to the Lebesgue measure on $\mathbb{R}$, i.e.,
$$m^*(A)=\inf\{\sum_{n=1}^\infty l(I_n):A\subset\bigcup_{n=1}^\infty I_n\},$$
where $A\subset\mathbb{R}$, $I_n\subset\mathbb{R}$ is a bounded open interval for $n=1,2,\dots$ and $l((a,b))$ is the length of the interval $(a,b... |
Keywords
positive linear maps, geometric mean, sector matrix, norm inequality
Abstract
Ando proved that if $A, B$ are positive definite, then for any positive linear map $\Phi$, it holds \begin{eqnarray*} \Phi(A\sharp_\lambda B)\le \Phi(A)\sharp_\lambda \Phi(B), \end{eqnarray*} where $A\sharp_\lambda B$, $0\le\lambda\l... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message Michael Spivak Joined: 10 Oct 2005 Posts: 52
Posted: Tue Oct 11, 2005 3:46 pm Post subject: new version of fonts We are making a new version of the MTPro fonts, which will have Times-Italic-like characters designed into them, so that there will... |
I am working on using a Feedforward multi-layered perceptron as a function approximator for the pressure distribution of a groundwater system. I am essentially trying to solve a boundary value problem with an ANNs.
From the mass balance equation of groundwater flow I now that the pressure, P, is dependent on the positi... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message stubner Joined: 14 Mar 2006 Posts: 7
Posted: Wed Apr 19, 2006 2:19 pm Post subject: absolute values Hi everybody,
it seems I ahven't used much absolute values lately since only yesterday I found that things like $|x|$ or $|o|$ look offbalance t... |
Let $f:\mathbb{R}\rightarrow\bar{\mathbb{R}}$ Lebesgue integrable. Prove that for $\epsilon >0$ there exists a finite interval $[a,b]$ such that
$$\left|\int{f(x)}dx-\int_{a}^b f(x)dx\right|<\epsilon.$$
My attempt: If $f$ is integrable on $[a,b]$, then for any $\epsilon > 0$ there exists $\delta > 0$ such that for any ... |
PCTeX Talk Discussions on TeX, LaTeX, fonts, and typesetting
Author Message stubner Joined: 14 Mar 2006 Posts: 7
Posted: Wed Apr 19, 2006 2:19 pm Post subject: absolute values Hi everybody,
it seems I ahven't used much absolute values lately since only yesterday I found that things like $|x|$ or $|o|$ look offbalance t... |
The following are both plausible messages, but have a completely different meaning:SOS HELP = ...---... .... . .-.. .--. => ...---.........-...--.I AM HIS DATE = .. .- -- .... .. ... -.. .- - . => ...---.........-...--.
You don't need a separator because Huffman codes are prefix-free codes (also, unhelpfully, known as ... |
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Revista Brasileira de Cirurgia Cardiovascular, ISSN 0102-7638, 2015
Journal Article
2010, 1. ed., ISBN 9871172559, 228, [3]
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1991, Ensayo crítico, ISBN 9789506001742,... |
Table of Contents
The Direct Product of an Arbitrary Collection of Groups
Recall from The Direct Product of Two Groups page that if $(G, \cdot)$ and $(H, *)$ are groups then the direct product of these groups is another group, $G \times H$ with the operation defined for all $(g_1, h_1), (g_2, h_2) \in G \times H$ by:(1... |
The Factorization of Polynomials with Real Coefficients
Table of Contents
The Factorization of Polynomials with Real Coefficients
We are about to look at an important way to factor polynomials with real coefficients, but before we do, we must first look at the following proposition.
Proposition 1: A quadratic polynomia... |
The general solution of second-order Cauchy-Euler equation
$$x^2y''(x)+pxy'(x)+qy(x)=0\tag1$$
is given by
$$y(x)=c_1 x^{\alpha_1}+c_2 x^{\alpha_2},\tag2$$
where $$\alpha_{1,2}=\frac{1-p}2\pm\frac{\sqrt{(1-p)^2-4q}}2.\tag3$$
But when $q=\frac14(1-p)^2$, i.e. when $\alpha_1=\alpha_2=\alpha$, the general solution somehow ... |
So after the other answer and comment I really really hope you ment working with ListPlot. Otherwise im sad :D
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So it's not an anwser with a pretty short code but it works well.
To uniformly distribute your points, we want the length between adjacent points to be constant. So we formulate:
$$l=\sqrt{(\Delta x)^2+(\Del... |
The coordinates of an event in spacetime are given by the 4-vector $(ct, \mathbf{r})$, where $\mathbf{r}$ is the spacial coordinates of the event. This 4-vector can be seen as 4-displacement of a worldline from the defined origin of the reference frame we're in at time $t$.
It seems sensible that $\frac{d}{dt}(ct,\math... |
Expected number of substitutions
Let's assume a haploid mutation rate of $\mu$ and a population of constant size $N$. For simplicity, I will assume an absence of selection because otherwise, we would need to talk about what kind of selection you want to talk about.
There are therefore $2 N \mu t$ mutations since this l... |
Examples of Expressing Integers as a Sum of Two Squares
Recall from the Expressing Integers as a Sum of Two Squares page that if $n \in \mathbb{N}$ and $n$ has prime power factorization $n = 2^{\alpha}p_1^{e_1}p_2^{e_2}...p_k^{e_k}q_1^{f_1}q_2^{f_2}...q_l^{f_l}$ where $p_i \equiv 1 \pmod 4$ for each $1 \leq i \leq k$ a... |
Most of the algorithms for estimating the volume of a convex polyhedron $K \subset R^d$ assume the existence of an affine transform $T$ with the property that
$$ B \subset TK \tilde{\subset}\ \sigma B$$
where $B$ is the unit ball in $d$ dimensions, and $\sigma$ is $O(\sqrt{d})$.
(Update: the $\tilde{\subset}$ indicates... |
Skills to Develop
In this section, we strive to understand the ideas generated by the following important questions:
What is a sequence? What does it mean for a sequence to converge? What does it mean for a sequence to diverge?
We encounter sequences every day. Your monthly rent payments, the annual interest you earn o... |
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