Related papers: Is the tautochrone curve unique?
General relativity predicts the existence of closed timelike curves (CTCs), along which an object could travel to its own past. A consequence of CTCs is the failure of determinism, even for classical systems: one initial condition can…
We formulate a uniqueness conjecture for curve shortening flow of proper curves on certain symmetric surfaces and give an example of a non-flat metric on the plane with respect to which curve shortening flow is not unique. That is, with…
Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for essentially any quantum control…
Examples are given of the creation of closed timelike curves by choices of coordinate identifications. Following G\"odel's prescription, it is seen that flat spacetime can produce closed timelike curves with structure similar to that of…
A classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current…
Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be…
The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this question by utilising the quantum circuit…
The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function $A$ yield, for fixed temperature $T$, a curve in the plane $\mathbb{R}^2$ (one fluid) or a surface in 3-space $\mathbb{R}^3$…
We show that H\"{o}lder continuous incompressible Euler flows that satisfy the local energy inequality ("globally dissipative" solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of…
The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the…
We introduce the relativistic version of the well-known Henon's isochrone spherical models: static spherically symmetrical spacetimes in which all bounded trajectories are isochrone in Henon's sense, i.e., their radial periods do not depend…
We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G)…
We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraint in time and natural regularity assumptions. We provide first a notion of weak solution, inspired by the theory of curves of maximal slope, and…
Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…
We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of…
The (chordal) Loewner differential equation encodes certain curves in the half-plane (aka traces) by continuous real-valued driving functions. Not all curves are traces; the latter can be defined via a geometric condition called the local…
Solutions of the Cauchy problem for the wave equation on a non-globally hyperbolic spacetime, which contains closed timelike curves (time machines) are considered. It is proved, that there exists a solution of the Cauchy problem, it is…
Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…
Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently designed new static spherically symmetric (Bertrand)…
This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical…