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№ 9
All Issues On the Relation between Curvature, Diameter, and Volume of a Complete Riemannian Manifold Abstract
In this note, we prove that if
N is a compact totally geodesic submanifold of a complete Riemannian manifold M, g whose sectional curvature K satisfies the relation K ≥ k > 0, then \(d(m,N) \leqslant \frac{... |
LaTeX:Symbols
LaTeX About - Getting Started - Diagrams - Symbols - Downloads - Basics - Math - Examples - Pictures - Layout - Commands - Packages - Help
This article will provide a short list of commonly used LaTeX symbols.
Contents Common Symbols Operators Relations Finding Other Symbols
Here are some external resourc... |
SolidsWW Flash Applet Sample Problem 3
Line 290: Line 290:
Answer submission and checking is done within WeBWorK. The applet is intended to aid with visualization and is not used to evaluate the student submission.
Answer submission and checking is done within WeBWorK. The applet is intended to aid with visualization a... |
I suspect there is confusion about understanding the problem. The problem is really asking you whether you are allowed to conclude that $(a_n)$ is Cauchy if
all you know is the inequality as stated.
Thus, in order to solve (a) you must find a
particular concrete sequence which satisfies the condition but is not Cauchy.... |
For predicted labels $\hat{y}$ and true labels $y\in\{0,1\}$, the confusion matrix is given by
\begin{array}{c|c:c|c} & y=0 & y=1 & \\\hline\hat{y}=0 & \mathrm{TN} & \mathrm{FN} & \hat{\mathrm{N}} \\\hdashline\hat{y}=1 & \mathrm{FP} & \mathrm{TP} & \hat{\mathrm{P}} \\\hline& \mathrm{N} & \mathrm{P} & (n_{\mathrm{obs}})... |
Above, we defined velocity as the derivative of position and acceleration as the derivative of velocity. Integration allows us to go the other way! As you learned in first semester calculus, integration allows us to generate an object's velocity as a function of time given its acceleration and an initial velocity. We c... |
I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well. The particle is inside a box with domain $\Omega=(0,0)$X$(1,1)$. For this I have $\hbar = m = 1$ and the energy is given analytically by $E_{m,n}=\frac{\pi^2}{2}(n^2 + m^2)$
My approximation... |
Sometimes, you may end up having to calculate the volume of shapes that have cylindrical, conical, or spherical shapes and rather than evaluating such triple integrals in Cartesian coordinates, you can simplify the integrals by transforming the coordinates to cylindrical or spherical coordinates. For this topic, we wil... |
Nature's Ron Cowen reviewed a technical paper in Nature that is one month old, Remotely related: sci-fi gets real:tech junkies should look at 27 science-fiction concepts that morphed into reality in 2012. Writing and Deleting Single Magnetic Skyrmions (Niklas Romming and 7 co-authors from Hamburg).See also reviews in G... |
The answer is quite simple.
The correlation matrix is defined thus:
Let $X = [x_1, x_2, ..., x_n]$ be the $m\times n$ data matrix: $m$ observations, $n$ variables.
Define $X_b= [\frac{(x_1-\mu_1 e)}{s_1}, \frac{(x_2-\mu_2 e)}{s_2}, \frac{(x_3-\mu_3 e)}{s_3}, ...]$ as the matrix of normalized data, with $\mu_1$ being me... |
Defining parameters
Level: \( N \) = \( 27 = 3^{3} \) Weight: \( k \) = \( 3 \) Nonzero newspaces: \( 3 \) Newforms: \( 4 \) Sturm bound: \(162\) Trace bound: \(1\) Dimensions
The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(27))\).
Total New Old Modular forms 69 51 18 Cusp forms 39 35 ... |
Let $u,v\in W^{1,p}(\Omega )\cap L^\infty (\Omega )$, $p\in[1,\infty ]$. Then, $u,v\in W^{1,p}(\Omega )$ and $$\partial _i(uv)=u\partial _iv+v\partial _iu.$$
I have problem to understand the proof. Let $p\in [1,\infty )$ and let $D\subset \subset \Omega $ an open. Let $\rho_n$ ba a standard mollifier. Define for $n$ la... |
Kale, GM and Jacob, KT (1989)
Gibbs Energies of Formation of $CuYO_2$ and $Cu_2Y_2O_5$ and Phase Relations in the System Cu-Y-O. In: Chemistry of Materials, 1 (5). pp. 515-519.
PDF
2008-14.pdf
Restricted to Registered users only
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Abstract
Thermodynamic properties of cuprous and cupric y... |
I read on the arXiv the following:
Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let $\mathbf{\Gamma}$ be the set of isomorphism classes of simple objects.
Unfortunately I could not read further since I didn't understand the jargon:
category: a collection of objects with morp... |
On p. 76 of the 1996 edition of Serre's
A Course in Arithmetic, one reads the following (inline) remark:
One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and is equal to $k$.
Here, $A$ is a subset of $\bf P$ (the set of all positive rational primes), and the natural density of $A$ ... |
Consider a Lagrangian theory of fields $\phi^a(x)$. Sometime such a theory posseses a symmetry (let's talk about internal symmetries for simplicity), which means that the Lagrangian is invariant under replacement $\phi^a\to \phi'^a=\phi'^a(\phi,\epsilon)$. Here $\epsilon$ are some continuous transformation parameters. ... |
Ready to get your head cracked? Ok, let’s define a simple function that multiplies each number of a list of numbers by 2. We will give this function the name of “by2”.
So, you have a function that takes a list of numbers as a parameter, and after the computation process shows a list of numbers. How do you write this in... |
This is a long comment, not an answer.
Let $E_k$ be the total space of the orientable bundle over $S^2$ with fiber $\mathbb R^2$ and Euler class $k$. $E_k = \mathbb R^2 \rtimes_k S^2$. Let $\pi : E_k \to S^2$ be the bundle projection.
$C_2 E_k = \{ (x,y) \in E_k^2 : x \neq y\}$ is the configuration space, with $p : C_2... |
To send content items to your account,please confirm that you agree to abide by our usage policies.If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.Find out more about sending content to .
To send content items to your Kindle, first ensure no-rep... |
Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
Prove that if $(f_n)$ is a sequence of Borel measurable functions and if $f(x)=\lim_{n\to \infty}f_n(x)$ exists in $\mathbb{R}$, then $f$ is Borel measurable. In fact $f$ is Borel measurable even if we only have $f(x)=\lim_{n\to\infty}$ almost everywhere on $D$, some measurable domain.
Attempt/Thoughts:
Suppose $(f_n)$... |
This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so many ... |
There is no acceptable/viable mechanism for a free electron to absorb or emit energy, without violating energy or momentum conservation. So its wavefunction cannot collapse into becoming a particle, right? How do 2 free electrons repel each other then?
It is true that the reactions $$e + \gamma \to e, \quad e \to e + \... |
So I'm already aware of the quantum mechanical operator for momentum and how to derive the kinetic energy operator from this: $$\hat T=\frac{\hat p^2}{2m}=\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2}$$ But I'm wondering how to derive the kinetic energy operator solely from the statistical definition of an expecta... |
If we want to describe a static spherically symmetric star we can use a metric which matches the Schwarzschild solution with correct mass on the outside of the star but differs from Schwartzschild in the inside of the matter distribution.
Basically we solve the Einstein equations with a source $T_{\mu\nu}$, for instanc... |
Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.
For exercises 1 - 4, the cylindrical coordinates \( (r,θ,z)\) of a point are given. Find the rectangular coordinates \( (x,y,z)\) of the point.
1) \( (4,\frac{π}{6},3)\)
Answer: \( ... |
In a previous question, I was looking for an equation for counting the number of the number of integers between $1$ and $x$ that have a prime factor besides $2$ or $3$.
There were 2 iterative equations that came up:
$x−\left\lfloor{log_2x}\right\rfloor−\left\lfloor{log_2\frac{x}{3}}\right\rfloor−\left\lfloor{log_2\frac... |
Learning Outcomes
Find the union of two sets. Find the intersection of two sets. Combine unions intersections and complements.
All statistics classes include questions about probabilities involving the union and intersections of sets. In English, we use the words "Or", and "And" to describe these concepts. For example,... |
Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. This happened to me when thinking about non-Lagrangian descriptions of QFT's. Below I'll provide a brief description of my ... |
If I take the definition of David Aldous and Jim Fill, a finite state space Markov chain is time-reversible if it satisfies the detailed balance equation$$\pi_i\,p_{ij}=\pi_j\,p_{ji}$$where the $p_{ij}$'s are the terms of the Markov transition matrix and the $\pi_i$'s are the terms of a probability distribution. Then, ... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
I have been thinking about quotients lately and pondered the following:
Let $G$ be a connected linear algebraic group and $X$ a $G$-variety where the action is the morphism $\sigma:G\times X\rightarrow X$. Let $p:L\rightarrow X$ be a line bundle on $X$.
A $G$-linearisation of $L$ is an action of $G$ on $L$ such that $p... |
Reading Ravenel's "green book", I wonder about his question on p.15 "that the spectrum MU may be constructed somehow using formal group law theory without using complex manifolds or vector bundles. Perhaps the corresponding infinite loop space is the classifying space for some category defined in terms of formal group ... |
Three standard deviations mean that the null hypothesis doesn't seem "great". The deviation from the predictions isn't enough for a discovery in a hard scientific discipline such as particle physics. However, I am convinced that a 3-sigma deviation – formally equal to a 99.7% certainty of a new effect – simply has to b... |
Jha, Ramanand (1994)
A Cosmology without Big Bang. In: General Relativity and Gravitation, 26 (11). pp. 1067-1073.
PDF
A_Cosmology-119.pdf
Restricted to Registered users only
Download (281kB) | Request a copy
Abstract
In contrast to standard ECSK theory with theBrans-Dicke scalar field $(\Phi)$ nonminimally coupled to ... |
For a parametrically defined curve we had the definition of arc length. Since vector valued functions are parametrically defined curves in disguise, we have the same definition. We have the added benefit of notation with vector valued functions in that the square root of the sum of the squares of the derivatives is jus... |
EDIT based on comments below:I add the mathematical formulation of my problem below. I am trying to solve an equation of the form$$\partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv G(x,y,t),$$discretizing this equation we have$$f^{k+1}_{i,j}= f^k_{i,j} + \Delta t G^k_{i,j}$$where $i,j$ refer to discreti... |
55 1
I'm reading a book on AdS/CFT by Ammon and Erdmenger and chapter 3 covers supersymmetry. This isn't my first look at SUSY but it's my first in depth look to really try to understand it, and when they talk about constructing a Lagrangian for ##\mathcal{N}=1## chiral superfields they write the most general form,
$$\... |
Thank you for using the timer!We noticed you are actually not timing your practice. Click the START button first next time you use the timer.There are many benefits to timing your practice, including:
Does GMAT RC seem like an uphill battle? e-GMAT is conducting a free webinar to help you learn reading strategies that ... |
Let $k$ be the length of any edge of a regular tetrahedron.Show that the angle between any edge and a face not containing the edge is $\arccos(\frac{1}{\sqrt3})$.
Let the regular tetrahedron be $OABC$.Let $O$ be the origin and position vectors of $A,B,C$ are the $\vec{a},\vec{b},\vec{c}$.
Let us find the angle between ... |
Suppsoe we are given an integer $n$. Define \begin{align*} \psi \left( n \right) = \left| \left\{ a \in \mathbb{Z}/ n\mathbb{Z}^\times \vert a^{n-1} \neq 1\right\} \right| \end{align*} Show: if $\psi \left( n \right) \geq 1$, it holds that $\psi \left( n \right) \geq \frac{1}{2} \phi \left( n \right)$, where $\phi \lef... |
So we try by applying the definition and see how it goes.
So we deal with 2 cases, either $\sup S=\infty$ or $\sup S=M<\infty$.
If it is the second case, this means that for all $x\in S$, $x\leq M$, so naturally $ax\leq aM$. By definition of the least upper bound, we have $\sup aS\leq aM=a\sup S$.
For the reverse, let ... |
Does math
require an $\infty$? This assumes that all of math is somehow governed by a single set of universally agreed upon rules, such as whether infinity is a necessary concept or not. This is not the case.
I might claim that math does not
require anything, even though a mathematician requires many things (such as co... |
I was told some days ago that the possibility of two randomly picked numbers are relatively prime to each other is $6/(\pi^2)$. And it is well known that the value of Riemann zeta function at 2 is $(\pi^2)/6$. So I guess there is a correspondence between them. Maybe the possibility of $n$ randomly picked numbers are re... |
I am wondering how to find the eigenvalues of some sparse matrix in given interval [a, b] by iterative method. To my personal understanding, it is more obvious to use Krylov subspace method to find the extreme eigenvalues rather than the interior ones.
The following strategy is called
shift and invert and depends upon ... |
In the following picture why Ksp is not simply S2- ? Why is f included with one species only? Please explain the last part.
What does f represent and why it is used in molar solubility?
Chemistry Stack Exchange is a question and answer site for scientists, academics, teachers, and students in the field of chemistry. It... |
I'm learning to integrate and I'd like to hear what are you favorite integration tricks?
I can't contribute much to this thread, but I like the fact that:
$$\int_{-a}^{a}{f(x)}dx=0 \space\text{if}\space f(x) \space\text{is odd}$$
Mathematics Stack Exchange is a question and answer site for people studying math at any l... |
Solve for $-\pi <\theta < \pi$: $$\tan\theta=\cos\theta$$
I can't get to the correct solution using the identities: $$\tan\theta=\frac{\sin\theta}{\cos\theta} \quad\text{and}\quad \sin^2\theta+\cos^2\theta=1$$
The answer I'm getting is
$$\sin\theta=-\frac12\pm\frac12 \sqrt{5}$$
giving: $0.62$ and $-1.62$.
The answers i... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
Recall the theorem that says that if a first order differential satisfies continuity conditions, then the initial value problem will have a unique solution in some neighborhood of the initial value. More precisely,
Theorem: A Result For Nonlinear First Order Differential Equations
Let
\[ y'=f(x,y) \;\;\; y(x_0)=y_0 \]
... |
Let's say we have two waves moving along a string. One of them is represented by the function: $$f_1(t)=\sin(\omega t)$$
The other one is represented by a function:
$$f_2(t)=-\sin(\omega (\tau-t))$$
Both of these functions are defined over one period.
At time $t=\tau/2$, the waves are overlapping perfectly and destruct... |
Hello, how do I find dy/dx of Please work it out for me to see rather than just give an answer. Thanks!And also, since I'm still trying to figure out LaTex, will you show me how to display this equation using this coding?
Follow Math Help Forum on Facebook and Google+
Use the derivative for $\displaystyle y=a^x$ and us... |
I apologize if this question is trivial, but I am new to physics and am struggling with some of the basic concepts.
Working in $\mathbb{R}^2$ with standard coordinates $(x,y)$, suppose we have a particle of mass $m$ moving on a curve $(x(t),y(t))\in\mathbb{R}^2$. It's tangent vector (velocity vector) is $$x^\prime(t)\f... |
There's a bit more to the story. Mathematica treats variables as complex by default, and I for one have had trouble figuring out how
Limit figures out how to treat variables such as
c in this case.
Some analysis
First, let's examine
a0 (
= a in OP) with the assumption that
c is real:
a0 = (h^2 + c^2 h^2 + Sqrt[4 h^2 + ... |
Posted by Nate on August 24, 2017 A Data-Driven Approach to LaTeX Autocomplete Autocomplete
Nearly anywhere you go on the web today you will find some sort of autocomplete feature. Start typing into Google and you get immediate suggestions related to your query. If you code in other languages, many IDEs have built-in, ... |
Yes, LED dimming can be done with constant current drivers and can even be done with that particular chip. However, you will need additional circuitry to achieve it.
To imagine what's needed and how, think about how LED PWM control is done professionally.
A constant current driver set to a specific current value in ord... |
Measurement of diffractive photoproduction of vector mesons at large momentum transfer at HERA 92 Downloads Citations Abstract.
Elastic and proton–dissociative photoproduction of \(\rho^0\), \(\phi\) and \(J/\psi\) vector mesons (\(\gamma p\rightarrow Vp\), \(\gamma p\rightarrow VN\), respectively) have been measured i... |
Learning Objectives
Identify a conic in polar form. Graph the polar equations of conics. Define conics in terms of a focus and a directrix.
Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, a... |
What is the Jacobian matrix?
What are its applications?
What is its physical and geometrical meaning?
Can someone please explain with examples?
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 up.Sign up to... |
You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for the coefficients.
A general matrix in the $SU(2)$ group is written as$$ M = \exp[ i( \alpha J_+ + \bar\alpha J_- + \gam... |
I wish to solve an equation of the form,
$$ \frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F}) $$
for the variable $\phi$ (e.g. mass).
On the right-hand side is the flux $\mathcal{F}$ of quantity $\phi$.
This equation
"looks like" an advection-diffusi... |
I'm studying about the finite element method in a class but I don't come from a civil engineering background. Anyways, it hasn't been made clear to me what the difference between constitutive laws and governing equations are. To me they both relate physical quantities with one another.
A constitutive law is generally a... |
My question is regarding Andreev bound states and their transmission probabilities. But to make this self-contained, lets quickly recap, for which I will draw from Tosi, L., Metzger, C., Goffman, M. F., Urbina, C., Pothier, H., Park, S.,Krogstrup, P. (2018). Spin-orbit splitting of Andreev states revealed by microwave ... |
In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. But I see in the literature that they write $Vir\otimes \overline{Vir}$ instead. The same happens in the more general ca... |
Assume I have two functions $f$ and $g$, with derivatives of $g$ at point $x$ and derivatives of $f$ at point $g(x)$ available.
What is the fastest way of computing derivatives of $f \circ g (x)$ ?
Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific prob... |
Proving that $$\sum_{n=0}^{\infty }\frac{1}{(2n)!!}=\sqrt{e}$$ Firstly, I tried to check the value with the exponential function at $x=.5$ but I found its terms not equal to the series terms.
Note that $$(2n)!! = 2\cdot4\cdot 6 \cdots 2n = 2^n n!$$ so your series is just $$\sum_n \frac{(1/2)^n}{n!} = e^{\frac{1}{2}}$$
... |
In the context of Noether's theorem , the Hamiltonian is the constant of motion associated with the time-translational invariance of the Lagrangian. Time-translational invariance is equivalent to the Lagrangian not depending explicitly on time that is $$\dfrac{∂L}{∂t}=0 .$$
The reason they're equivalent is that for an ... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
I have to solve the Klein Gordon equation for a scalar field, in global $AdS_3$ (covering space, with non periodic $\tau$) written as $$ ds^2=\frac{R^2}{\cos^2\rho}(-d\tau^2+d\rho^2+\sin^2\rho d\theta^2) $$ I Fourier transformed the scalar field as $$ \phi(\rho,\theta,\tau)=\sum_l\int d\omega Y_l(\cos\theta)e^{ik\omega... |
"A crystal is formed by a large number of repetitions of basic pattern of particles in space.
The basic structural unit which when repeated in three spatial directions generates the crystal structure is called unit cell."
Problem:
Assume I take some same perfect geometrical cubes and arrange them in all three dimension... |
Note that by denoting $f(x) = \tan x \sin x -x^2$, you found that $$f'''(x)=-\sin x (1-6\sec^4x+\sec^2x) = \sin x (1+3\sec^2 x)(2\sec^2 x -1 ) \geq 0 $$Hence $f''(x)$ is increasing, with $f''(0)=0$, we conclude that $f''(x) \geq 0$.
Hence $f'(x)$ is increasing, with $f'(0)=0$, we conclude that $f'(x) \geq 0$.
Hence $f(... |
I am trying to understand the derivation of the following equation which describes the motion of Newtonian viscous fluids:
Equation 1: $\rho \dfrac{D u}{Dt} = \rho g_x - \dfrac{\partial p}{\partial x} + \mu \nabla^2 u$
Where $\mu$ is the viscosity of the fluid. I am following the proof written in Hibbeler's Fluid Dynam... |
I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone review the following, is it correct?
The finite-difference matrix
The Poisson equation,
$$ \frac{\partial^2u(x)}{\parti... |
Let $p$ be a prime. Let $G$ be a solvable, non-regular, transitive permutation group such that some element fixes no point, and each element fixing some point fixes exactly $p$ points. Suppose that for $g \notin N_G(G_{\alpha})$ we have $$ G_{\alpha}^g \cap G_{\alpha} = 1 $$ and that $p$ does not divide the order of $G... |
Let $(\Omega, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $g$ a measureable function on $\Omega$. Fix $p\in[1,\infty]$ and consider the multiplication operator $$M_g:L^p(\Omega, \mu)\to L^p(\Omega, \mu),\quad f\mapsto fg$$ Then $M\in B(L^p(\Omega, \mu))$ if and only if $g \in L^\infty(\Omega, \mu)$ and in ... |
Two parts of this answer: (1) a quick sketch of how one gets the result cited and (2) why I believe one can't get a less awkward, more succinct result.
Proof Sketch
The one thing that the numerical version of Snell's law does not give us, but which we always unconsciously use, is the equally important fact that the inc... |
Čech cohomology fails for the plane with the doubled origin, $\mathbb A^2_{00}=\mathbb A^2\cup_{\mathbb A^2-0}\mathbb A^2$, with cohomology in the structure sheaf: $\check{H}^2(\mathbb A^2_{00};\mathcal O)=0\ne H^2(\mathbb A^2_{00};\mathcal O)$. Čech cohomology fails for both the Zariski and étale cohomologies.
Mayer-V... |
501 0
In the vortex panel method the following equation is used
[tex] V_{freestream}sin \beta_i - \sum_{j=1}^n \displaystyle\frac{\lambda_i}{2\pi} \int \displaystyle\frac{d\theta_{ij}}{dn_i} ds_j = 0[/tex]
where n is the panel number, i is the control point at which the vortex strength is being calculated and j is the ... |
Statistical properties of stochastic 2D Navier-Stokes equations from linear models
1.
University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071
2.
Università di Pavia, Dipartimento di Matematica, via Ferrata 5, 27100 Pavia, Italy
In this paper, we investigate this conj... |
I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?
I have so many ideas for this ...... |
I was trying to solve this 1D diffusion problem \begin{equation} \dfrac{\partial^2 T}{\partial \xi^2} = \dfrac{1}{\kappa_S}\dfrac{\partial T}{\partial t}\, , \label{eq_diff_xi} \end{equation} with the boundary conditions \begin{align} &T(\xi = 2Bt^{1/2},t) = A t^{1/2}\, ,\\ &T(\xi=\infty,t) = 0\, ,\\ &T(\xi,0) = 0\, , ... |
The theoretical tools at my disposal are Abel's test and Dirichlet's test. To recap those:Say I have an integral of the form $$\int_{a}^{b}f\cdot g \hspace{1.5mm} dx$$with Improperness (vertical or horizontal asymptote) at
b.
Abel's test guarantees convergence for
$\bullet$ $g$ monotone and bounded on $(a,b)$ $\hspace{... |
This post is a sequel to Concatenate and average NetCDFs where I introduced NCO (NetCDF operators), how to install NCO, and provided examples for the essential operations of record and ensemble concatenation and averaging of variables across multiple NetCDF files. If you are new to NCO I suggest familiarizing yourself ... |
Is there any hope in solving the following linear system efficiently with an iterative method?
$A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^n \text{, with } n > 10^6$
$Ax=b$
with
$ A=(\Delta - K) $, where $\Delta$ is a very sparse matrix with a few diagonals, arising from the discretization of t... |
For $z > 0$, you have $B = - \sinh(ka) / \cosh (ka)$, so
$$ \phi = \frac{A}{\cosh (ka)} \left( \cosh (ka) \sinh (kz) - \sinh(ka) \cosh(kz) \right) = \frac{A}{\cosh(ka)} \sinh(k(z-a)).$$
[By the way, if you had written the general solution in the form $\phi = C \sinh(k(z - a)) + D \cosh (k(z - a))$, then it would have b... |
I am solving this for particles distributed in a 2D space. First we know that the moment of inertia of a particle about an axis is given by
$$I=Mr^2$$
And we know that the axes passing through COM have the minimum moment of inertia. Our job is to find the slope of the axes with min Inertia, then we can use the slope po... |
Do you want iPad2 or iPod Touch for free?
The prize for KAIST Math POW is getting better, thanks to the department support. From this Fall semester, we will have the following as the prizes:
1st prize: iPad2 16GB
2nd prize: iPod Touch 32GB
3rd prize: 5 WEEKDAY DINNER gift certificates for for a buffet restaurant in Yus... |
36 0 1. Homework Statement
Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## , and which satisfies... |
Let $\{X_\alpha\}_{\alpha \in I}$ be a collection of mutually disjoint measurable subsets of $\mathbb{R}. Show that at most countable of them has positive measure.
I want to see if my proof is correct.
Proof:
Let $\{X_\alpha\}_{\alpha \in \Gamma \subset I}$ be the subcollection such that $m(X_\alpha) > 0, \forall \alph... |
Consider a test particle of mass $m$ which is in orbit around a spherical-symmetric body with mass $M$. It therefore has a position as described by the coordinates $r,\phi$, and its motion can be described by the Lagrangian $L$ of the Einstein-Infeld-Hoffman-Equations:
$$L = \frac{mv^2}{2}+ \frac{GmM}{r}+\frac{mv^4}{8c... |
Attractors¶ Visualizing Attractors¶
An attractor is a set of values to which a numerical system tends to evolve. An attractor is called a strange attractor if the resulting pattern has a fractal structure. This notebook shows how to calculate and plot two-dimensional attractors of a variety of types, using code and par... |
Suppose $A$ is a Borel measurable subset of [0,1], $m$ is Lebesgue measure, and $\varepsilon\in (0,1)$. Prove that there exists a continuous function $f: [0,1]\to \mathbb{R}$ such that $0\le f\le 1$ and $$ m(\{x:f(x)\ne\chi_A(x)\})<\varepsilon. $$ Here $\chi_A(x)$ is the indicator function on $A$.
If $A$ is Borel measu... |
I am assuming a very simple case, where there is only a mass $m$ with position $x$ under an external force $F$. we know that the Lagrangian takes the form $L = (1/2) m \dot{x}^2$ from which equations of motion follow as $$\frac{d}{d t} \frac{\partial L}{\partial \dot{x}}= m \ddot{x} = F\tag{1}$$ respectively.
Now, cons... |
I'll begin with a general remark: first-order information (i.e., using only gradients, which encode slope) can only give you directional information: It can tell you that the function value decreases in the search direction, but not for how long. To decide how far to go along the search direction, you
need extra inform... |
It's well-known that $\sin(\pi \frac pq)$ is always algebraic. In particular, as I understand, it can always be expressed in terms of radicals, because it can be connected to the abelian group of $e^{i\pi \frac pq}$. Because these can always be expressed in terms of radicals -- this means there must be some other algeb... |
This question already has an answer here:
Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals... |
Derivatives of PGF of Bernoulli Distribution Theorem $\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \begin{cases} p & : k = 1 \\ 0 & : k > 1 \end{cases}$ $\Pi_X \left({s}\right) = q + ps$
where $q = 1 - p$.
We have that for a given Bernoulli distribution, $p$ and $q$ are constant. $\dfrac {\mathrm d} {\... |
I only have a partial answer for 1. and a hopefully non-confusing answer to 2.
To start with, let us work with the fundamental groupoid, which is more, ahem, fundamental and better suited to generalisation. In particular, we can consider the set $\pi^J(X,a,b)$ of homotopy classes (rel endpoints) of maps $(J,0,1) \to (X... |
Let $M$ be a smooth manifold and denote $C^\infty_0(M)$ the space of smooth functions with compact support. In Mathematics a
distribution is defined to be a continuous linear functional $\phi : C^\infty_0(M)\to \mathbb{R}$. The space of distributions is usually denoted $\mathfrak{D}'(M)$.
So a distribution is a map tha... |
1. Rational Functions (Definition)
Definition: Rational Function
A
rational function is a quotient of polynomials \(\dfrac{P(x)}{Q(x)}\).
Example 1
\[\dfrac{(x^2 + x - 1)}{(3x^3+ 1)},\]
\[\dfrac{(x - 1)}{(x^2 +1)}, \text{ and}\]
\[\dfrac{x^2}{(x + 1)}\]
are all Rational Functions
Example 2
Find the domain of
\[\dfrac{(... |
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