منابع مشابه
On Three Circles
Abstract. The classical Three-Circle Problem of Apollonius requires the construction of a fourth circle tangent to three given circles in the Euclidean plane. For circles in general position this may admit as many as eight solutions or even no solutions at all. Clearly, an “experimental” approach is unlikely to solve the problem, but, surprisingly, it leads to a more general theorem. Here we co...
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In this article we consider the following old Japanese geometry problem (see Figure 1), whose statement in [1, p. 39] is missing the condition that two of the vertices are the opposite ends of a diameter. (The authors implicitly correct the omission in the proof they provide on page 118.) We denote by O(r) the circle with centre O, radius r. Problem [1, Example 3.2]. The squares ACBD and ABCD h...
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We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constant function zero is the only solution that vanishes at infinity. Clearly, for ...
متن کاملApollonian Circle Packings
Figure 1: An Apollonian Circle Packing Apollonius’s Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes’ Theorem is if the initial four tangent circles have integr...
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The total entropy production of a trajectory can be split into an adiabatic and a nonadiabatic contribution, deriving, respectively, from the breaking of detailed balance via nonequilibrium boundary conditions or by external driving. We show that each of them, the total, the adiabatic, and the nonadiabatic trajectory entropy, separately satisfies a detailed fluctuation theorem.
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ژورنال
عنوان ژورنال: Proceedings of the Edinburgh Mathematical Society
سال: 1884
ISSN: 0013-0915,1464-3839
DOI: 10.1017/s0013091500037287