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A tautochrone curve or isochrone curve (from Ancient Greek ταὐτό ' same ' ἴσος ' equal ' and χρόνος ' time ') is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
A curve completely within the interior of another is a subset of it. Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2 n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by ...
The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve).
If the solution to any problem can be formulated recursively using the solution to its sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table (often an array or hashtable in practice). Whenever we attempt to solve a new sub-problem, we first check the table to see ...
Therefore, the computation of F(n − 2) is reused, and the Fibonacci sequence thus exhibits overlapping subproblems. A naive recursive approach to such a problem generally fails due to an exponential complexity. If the problem also shares an optimal substructure property, dynamic programming is a good way to work it out.
For instance, an object located at position p at time t 0 can only move to locations within p + c(t 1 − t 0) by time t 1. This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of t {\displaystyle t} for time and ct for space.
Here t is the time, G is the modulus, and T 0 < T 1 < T 2. Temperature dependence of elastic modulus of a viscoelastic material under periodic excitation. The frequency is ω, G' is the elastic modulus, and T 0 < T 1 < T 2. The time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids ...