"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Trajectory Code Problem

Problem 2. “Pumpkin chucking” is a competition event to see which team can shoot a pumpkin as far as possible, usually with a pneumatic cannon. In this problem we’re going to write a computer program ...
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67 views

Could you offer another way to prove $e^{\hat{A}}\hat{B}e^{-\hat{A}}=e^{ad\hat{A}}\hat{B}$

My professor wants me to solve this identity in two ways. Sadly, I could only do one way and haven't figure out how to solve it another way. Here is my way, Denote ...
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53 views

Punctual Hilbert scheme of four points

I am looking at $\text{Hilb}^4(\mathbb{C}^2)$, which is the Hilbert scheme of four points on $\mathbb{C}^2$. In particular, I am just looking at four points collided (at the origin), and want to know ...
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41 views

Notation in Reed/Simon Vol. IV (and possibly an earlier volume)

I'm wondering if there are any mathematical physicists/analysts out there that can help me with some notation I've seen in Reed and Simon's books on analysis. Unfortunately I don't have time to read ...
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57 views

FLATLAND's sphere intersection scenario, explored for four dimmensions

I recently finished this wonderful new vintage edition of FLATLAND. http://amzn.com/918775116X In 1884, Edwin Abbott wrote this strange and enchanting novella called FLATLAND, in which a square who ...
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Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
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60 views

Personal Experiences with Probability Simulation

Simulations methods are increasingly used in theoretical and (especially) applied probability. Personally, I have used simulation for purposes that range from recreational Q&A to applications of ...
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30 views

Mean-pressure of an acoustic wave

I am looking at total internal reflection for an acoustic wave, defined in terms of its pressure such that $$p = p_1 \,exp\left[-i\{\omega t-\vec{k} \cdot\vec x\}\right]$$ Using the definition of ...
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119 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification ...
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86 views

How can we proof that this is equal? About $ln(n)$

I found this on this website (Euler-Mascheroni constant expression, further simplification) without any explaining why this is equal can someone give me that? ...
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36 views

Green's function in the context of classical mechanics

I am following this paper entitled "The classical mechanics of non-conservative systems". I would like to discuss equation (2) since I cannot get what the autor says. This is the problem: let's ...
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30 views

Parameters in the Hamilton-Jacobi Equation

I'm reading through Gelfand and Fomin's 'Calculus of Variations', and they've just derived the Hamilton-Jacobi Equation: $$\frac{\partial S}{\partial x} + H \left(x, y_1, \ldots, y_n , \frac{\partial ...
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48 views

maximum position uncertainty of particle in a box

I want to verify mathematically for wave function $\psi(x)$ satisfying $\psi(x)=0$ for $\lvert x \rvert \ge \frac{L}{2} $ and $\int_{- \frac{L}{2}}^{\frac{L}{2}} \lvert \psi(x) \rvert ^2 dx = 1 $ ...
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94 views

What is the prerequisite knowledge for Navier–Stokes Existence and Smoothness problem?

I am highly interested in the Millennium Problem of Navier–Stokes Existence and Smoothness (also here) and my aim is to reach some level of knowledge to do research on it. The problem seems simple to ...
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75 views

Finding initial value of differential equation

Given, $$ mdv/dt = mg - kv $$ Question is: Find the velocity $v(t)$ that satisfies this initial value problem. Also, by letting $t$ approach positive infinity, determine the terminal velocity ...
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39 views

Blow-Up for Semi-Linear Wave Equation

I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
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43 views

Kahler-Einstein Metrics in Physics - Topic Suggestions

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ...
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86 views

Regge symmetry and outer automorphisms of Dynkin diagrams

Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has ...
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51 views

Function $\int_0^1 x^{a}(1-x)^{n}~dx$ used for Gamma Function

I was reading a historical note on Euler and found that below given function is used to find Gamma Function: $$ \int_0^1 x^{a}(1-x)^{n} dx .$$ And I could not understand that why this function ...
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97 views

What are BesselJ functions?

I solved an integration on mathematica which gives BesselJ functions and some other terms. I explored mathematica help and google but could not understand the difference between different types of ...
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37 views

partial differential equation applicational problem

As a Maths student with not much knowledge in physics, I dont understand how the "string" can be "cut" into half at x=L/2. Also, how many initial conditions(data) does this question have apart from ...
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45 views

How invariance is formulated mathematically?

Consider $M$ a smooth manifold of dimension $n$, then a vector at the point $a\in M$ can be defined without any reference to any coordinate system. Indeed, we define a vector $v\in T_aM $ usually as ...
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121 views

What a reference frame is mathematically?

Physicists usually talk about reference frames and more specially inertial reference frames. This is particularly important in Mechanics and Relativity. Now, from the Physics standpoint there's no ...
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76 views

For Riemann Hypothesis, many people seek physics intuition, why not for Goldbach Conjecture ?

All: As we all know, for Riemann Hypothesis research, many people seek physics intuition, to understand more fundamental reasons why Riemann Hypothesis shall hold. In this direction, we have ...
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Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?

I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while). Specifically what I'm ...
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32 views

Thermodynamics: find exit temp and velocity of air out of a nozzle?

I don't know if I can really ask a thermodynamics question here on this math site but I need help and this was the best site for when I needed help in math class. Concerning thermodynamics, I have ...
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2answers
73 views

How can I scale a value from -255 to 255 or -100 to 100 to a scale of 0-100?

For Brightness, I have a formula that takes in a value from -255 to 255 and contrast from -100 to 100. What if I wanted to use the same formula but I wanted to convert/adjust the scaling so that I ...
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1answer
68 views

Vectors Velocity, Physics Word Problem.

A bird is flying from Hamilton ON to Waterloo ON. There is a heavy wind traveling at 5.0km/h (S11°E). What should its heading be? How long will it take? That is all the information I get and ...
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26 views

Analytic solution to wave equation on a hollow cylinder

Is it possible to find an analytic solution for the modes of vibration of a hollow cylinder, assuming azimuthal symmetry? That is can the following PDE be evaluated: ...
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1answer
94 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
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2answers
82 views

Notion of contraction in tensor algebra

Assuming a vector space V and it's basis set $\{\vec{e}_\nu\}$. A vector $\vec{v}$ can be written as: $\vec{v}=x^\nu\vec{e}_\nu$ where $x^\nu$ is the corresponding contravariant coordinate. We can ...
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63 views

Can an integral of a function that is not well behaved be finite?

Consider the following integral which gives the time period of simple pendulum where $\theta_0$ is the initial inclination of pendulum with vertical. ...
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90 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
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55 views

Understanding derivatives in simple terms

Im am trying to understand the idea of derivatives and how they relate to the real world. I understand if i have function, in pkysics first derivative is the velocity, and the second derivative is ...
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172 views

Stefan-Boltzmann Constant and Stefan's Law

The following argument is from my textbook, An Introduction to Thermal Physics by Daniel Schroeder. If you are familiar with the derivation of Stefan's Law from the energy density of a photon gas, you ...
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70 views

Index notation confusion in tensor algebra

I have some confusions regarding index notation in tensor algebra. Let's assume $\vec{v}$ is a vector belonging to vector space $V$. Choosing a basis set $\{\vec{e}_\nu\}$, $\vec{v}=x^\nu\vec{e}_\nu$ ...
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37 views

Flow through a viscous fluid, “reflections method”

does anyone know about the "reflections method" on determining the velocity field around a small assemblage of spheres moving through a viscous fluid with small Reynolds Number? [ch. 6 Happel & ...
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59 views

complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines We need to take the value of m=1. I have an idea on how to do the parts and i know what a ...
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22 views

Modelling: Use Newton's law to write down an equation for the position $x$ of the mass.

Here is the background for the question: Consider a one-dimensional frictionless spring-mass system, where the forces acting on the mass $m$ at position $x$ are the forces of gravity $F_g =-mg$ with ...
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45 views

Justifying the use of real numbers for measuring length

I am not sure if this is the most appropriate place to post this but here goes nothing: Assume we were trying to come up with system of numbers $S$ to model our intuition of length. We want $S$ to ...
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22 views

How do you calculate certain variables of two or more events that occur simultaneously compared to the same events happening subsequently.

Say you have two hoses, A and B, that fill up a pool of equal size at different rates. Hose A fills up a pool in 10 mins, hose B in 20 mins. Thus A = 1p/10m, B = 1p/20m. Lets say that Hose A filling ...
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142 views

Generalized Functions (Distributions) over Manifolds

What is the right way of making sense of generalized functions over manifolds? For concreteness, let me restrict my question to the dirac delta function. The article on Wikipedia on Dirac delta ...
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15 views

Show that the current in a cylindrical conductor is uniformly distributed across its cross section

Using relevant equations for E and J, show that the current in a steady current I in a cylindrical conductor with uniform conductivity $\sigma$ is uniformly distributed across its cross-section. I ...
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25 views

Interpretation of integral.

The height, in centimeters, of a bicycle pedal is given by $h(t)=30+16\sin t$ where $t$ is the time. Evaluate and interpret the following integral \begin{align} \dfrac{1}{2\pi}\int_0^{2\pi} h(t)\,dt. ...
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14 views

Calculating gain ratio from a dB value

In a practice problem I have: power gain = $\log_{10}(\frac{db}{20})$ The final answer for the ratio is 1. The dB value is $-3$. When I do $\log_{10}(\frac{3}{20})$ I get $-0.823$. Just wondering ...
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46 views

find travel time given path and velocity field

As I was studying refraction, I began wondering what path would light take when entering a non-homogeneous transparent medium, i.e. a certain material in which the refraction index $n$ varies ...
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81 views

Completeness of solutions and the separation of variables method

The method of separation of variables is introduced in every textbook on mathematical physics. A basic question is rarely addressed: does this method exhaust all the solutions? Is there any ...
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86 views

Multiple absorbing boundaries

I am interested in the relation between absorbing boundaries and the trajectories of particles (evolving according to a Brownian motion). The probability to hit a boundary at a given time can be ...
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1answer
45 views

Classification of All Maps $T^2 \to S^4$ up to homotopy

From my study of physics I have arrived at the question of how to classify all maps $\mathbb{T}^2\to S^4$ where $\mathbb{T}^2$ is the two-torus. The classification should be up to homotopies. The ...
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40 views

Sturm-Liouville and Bessel function identity

Given S-L equation $\dfrac{1}{x}[\dfrac{d}{dx}(xy')+(\dfrac{-m^2}{x})y]=-\lambda y$ Say $\mathcal{L}$ is the Sturm-Liouville operator, $y_k$ is eigenfunction $J_m(j_{mk}x)$ where $J_m$ is Bessel ...