# Tagged Questions

"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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### Special case of the inverse Ising problem with equal correlations

Let $s_1,\dots,s_N\in \{-1,1\}$ be $N$ binary spins. The problem of finding a symmetric interaction matrix $J=(J_{i,j})_{i,j=1}^N$ with zero diagonal and an external magnetic field $h=(h_i)_{i=1}^N$ ...
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### How does the Pauli principle work?

Let $H$ be some Hilbert space. Now in general, in quantum mechanics, the vector space representing states of $n$ (non-interacting) particles is $H^{\otimes n}$, but if I consider these particles of be ...
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### Find the friction constant minimizing the duration of the vertical movement of a wheel

Q) The mass of a car that acts on one wheel is $100 kg$. The elasticity (spring) constant in the suspension system of that wheel is $k = 10^{4}N/m$. Design the strut (find the ...
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### Free Particle on Riemannian Manifold

I'm trying to understand Takhtajan's "QM for Mathematicians" and I'm struggling still with the generalized coordinates. To make things simple consider a free particle on some Riemannian manifold. ...
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### Calculate the line integral (cyl. coords)

So I have this vector field $$\textbf{B}=K \left( \frac{\cos \varphi}{\rho^2}\textbf{e}_{\rho}+ \left( \frac{\sin \varphi}{\rho^2}+ \frac{1}{a\rho}\textbf{e}_{\varphi} \right) \right)$$ and the ...
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### Simple (?) tensor index notation; When do the indices mean inner product and in what order?

In index notation, does the term $σ_{ik}x_{j}n_{k}$ mean $\bf{σx}\cdot\bf{n}$ or $\bf{xσ}\cdot\bf{n}$? Here $σ$ is a second-order tensor and $x,n$ are vectors. On the same note, is ...
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### What does symmetry imply about the solution in mathematics? (Example: Gauss' law)

Suppose you have an infinite cylinder and are considering a field $\mathbf{D}$ caused by physical elements within the cylinder such that it satisfies $\int \mathbf{D}\cdot d\mathbf{a} = Q_{free}$. ...
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### Cohomology of Anti-de Sitter manifold and black hole

Anti-de Sitter manifold AdS$_n$ is a maximally symmetric pseudo-Riemannian manifold with constant negative scalar curvature. This has $\mathbb{R}^{2,n-1}$ as its embedding and is a solution to the ...
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### What exactly does the Hilbert scheme of points parametrize?

The Hilbert scheme of points is defined as $$\text{Hilb}^n(X) = \{ I \subset \mathbb{C}[x,y] \text{ such that } \text{dim}_{\mathbb{C}}/I = n \}$$ or, in words, the Hilbert scheme of points ...
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### Rounding in the method of least squares for linear regression analysis?

I tend to round off the intermediate values (like that for $\Sigma x^2)$ of my calculation for slope, etc., to appropriate significant digits by considering the significance of the raw data. ...
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### How to plot a qubit on the Bloch sphere?

I've been reading pages such as this one: http://comp.uark.edu/~jgeabana/blochapps/bloch.html Which talk about the Bloch sphere, but I've been unable to figure out how to plot states on the sphere ...
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### What are Einstein's evolution equations for galaxies? [closed]

I'm researching galaxy distributions and have been tasked with solving Einstein's evolution equations for different levels of dark energy and matter. I've been told to do this numerically via Matlab, ...
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### Determine if this vector field is a conservative force field?

Does $A(x, \, y) = (3x^2 + 2y^2)i + (4xy + 6y^2)j$ represent a conservative force field? If so, determine the potential $\phi$ in $A = \text{grad} \ \phi.$ From what I understand, we need to ...
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### Solve this equation by a series of steps

Solve the equations for $v_x$ and $v_y$ : $$m\frac{d({v_x)}}{dt} = qv_yB \qquad m\frac{d{(v_y)}}{dt} = -qv_xB$$ by differentiating them with respect to time to obtain two equations of the ...
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### Why does for the $16$ representation of $O(10)$ the term $16 \otimes 16 \otimes 16 \otimes 16$ vanish?

Given the $16$ dimensional representation of the Lie group $O(10)$, why does the quartic term $$16 \otimes 16 \otimes 16 \otimes 16$$ vanish? Surely I'm missing something very obvious, because ...
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Given a representation $R$ of a group $G$ and the corresponding tensor product $$R \otimes R = R_1 \oplus R_2 \oplus R_3 \oplus \ldots$$ how can I compute how many quartic terms $\mathrm{O} (R^4) ... 2answers 156 views ### Is$\int |x\rangle \langle x|dx$Mathematical? I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ... 1answer 29 views ### Work against friction is proportional to length of path If, given that the frictional force is constant, one wants to show that the work done against friction is proportional to the length of the path, would this line of reasoning be correct? We can use ... 1answer 31 views ### Boundary value problem$y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$A physical problem brought me to the following boundary-value problem$y''(x)= \kappa^2 \left(y(x) + \frac{y(x)^2}{2} \right)$with$y(0)=0$and$y(C)=-58$for some$C>0.$If there was no ... 1answer 76 views ### I can't find any formula to solve this differential equation. $$\frac{dx}{dt} + x^2 = B + A\cdot e^{C\ln\big(\frac{x}{x_0}\big)+\ln(x_0)}, \quad x(t_0)=x_0$$ Can anyone please help me where I can start from this equation? I simplified a complicated equation ... 1answer 49 views ### How to rigorously understand continuous bases? In Quantum Mechanics it is quite common to see the idea of a continuous basis of a Hilbert space. In truth if$\mathcal{H}$is the state space of a quantum system and if$X : U\subset \mathcal{H}\to ...
Consider a system where a projectile is shot up a ramp. Let the ramp be inclined at some angle $\alpha$ and the projectile is shot at some angle $\theta > \alpha$ with fixed velocity $V$. If we ...