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Help. The free encyclopedia. Necklace splitting problem. In mathematics, and in particular combinatorics, the necklace splitting problem arises in a variety of contexts including exact division; its picturesque name is due to mathematicians Noga Alon [1] and Douglas B. West.[2] Suppose a necklace, open at the clasp, has k ·n beads. There are k · ai beads of colour i, where 1 ≤ i ≤ t. Then the necklace splitting problem is to find a partition of the necklace into k parts (not necessarily contiguous), each of which has exactly ai beads of colour i; such a split is called a k-split. The size of the split is the number of cuts that are needed to separate the parts (the opening at the clasp is not included).
Inevitably, one interesting question is to find a split of minimal size. Example of necklace splitting with k = 2 (i.e. two thieves), and t = 2 (i.e. two types of beads, here 8 red and 6 green). Alon explains that Further results[edit] One cut fewer than needed[edit] , a (t − 1)-split exists such that: Splitting multidimensional necklaces[edit] Boiling. Boiling is the rapid vaporization of a liquid, which occurs when a liquid is heated to its boiling point, the temperature at which the vapor pressure of the liquid is equal to the pressure exerted on the liquid by the surrounding environmental pressure.
Types[edit] Nucleate[edit] Nucleate boiling of water over a kitchen stove burner Nucleate boiling is characterized by the growth of bubbles or pops on a heated surface, which rise from discrete points on a surface, whose temperature is only slightly above the liquid’s. In general, the number of nucleation sites are increased by an increasing surface temperature. An irregular surface of the boiling vessel (i.e. increased surface roughness) or additives to the fluid (i.e. surfactants and/or nanoparticles[1]) can create additional nucleation sites,[2] while an exceptionally smooth surface, such as plastic, lends itself to superheating. Critical heat flux[edit] Transition[edit] Film[edit] In distillation, boiling is used in separating mixtures.
Leidenfrost effect. Leidenfrost droplet The Leidenfrost effect is a phenomenon in which a liquid, in near contact with a mass significantly hotter than the liquid's boiling point, produces an insulating vapor layer keeping that liquid from boiling rapidly. This is most commonly seen when cooking; one sprinkles drops of water in a pan to gauge its temperature: if the pan's temperature is at or above the Leidenfrost point, the water skitters across the pan and takes longer to evaporate than in a pan below the temperature of the Leidenfrost point (but still above boiling temperature).
The effect is also responsible for the ability of liquid nitrogen to skitter across floors. It has also been used in some potentially dangerous demonstrations, such as dipping a wet finger in molten lead[1] or blowing out a mouthful of liquid nitrogen, both enacted without injury to the demonstrator.[2] The latter is potentially lethal, particularly should one accidentally swallow the liquid nitrogen.[3] Effect[edit] Where, If. Elliptic function. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives.
Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms. Definition[edit] Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with (in other words, not parallel), such that for all , it follows that for any . C++ Reference [C++ Reference]
C++ Reference [C++ Reference]