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### Towards recognizing St. Venant geometrical invariant

Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates:
$$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...

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### Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks.
Consider the system consisting of the controlled object and regulator. The ...

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### Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...

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### Tying knots via gravity-assisted spaceship trajectories

Q.
Can every knot be realized as the trajectory of a spaceship
weaving among a finite number of fixed planets, subject to gravity alone?
To make this more ...

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### symplectic topology of (perturbed) KAM tori

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...

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### Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?

Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions:
Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...

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### How to make sense of the Euler Lagrange equations for an infinite action?

The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional
$S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$
Say we have an ...

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### Can one obtain this ODE as an Euler-Lagrange equation?

Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...

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### Conjecture: Finitely many points where gravitational field due to N masses vanishes

Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes. ...

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### A Stochastic Dynamical Billiard

Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle ...

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### Optimal contour shape for variational problem over captured area

Let's assume we have a continuous and finite scalar function $f(x,y)$ over the $xy$ plane ($\mathbb{R}^{2}$) and this function is to be integrated over a bounded area (surface) $A\subset\mathbb{R}^{2}...

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### Formulation of contour variational problem

I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer.
Within a bounded region on the $xy$ ...

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### Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions
are uniformly distributed in a unit-radius disk.
Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...

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### Do bubbles between plates approximate Voronoi diagrams?

For example, soap bubbles:
Image from UPenn:
"A 2-dimensional foam of wet soap bubbles squashed between glass plates, after 10 hours ...

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### Sampling point from the surface of an n-dimensional ellipsoid with uniform distribution

I am wondering if exist an efficient computational method for sampling points belonging to the surface of an ellipsoid in $n$-dimensional space with n even, I am thinking in the phase space of a ...