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Anti-concentration inequalities, on the other hand, provide an upper bound on how much a random variable can concentrate, either on a specific value or range of values. A concrete example is that if you flip a fair coin n {\displaystyle n} times, the probability that any given number of heads appears will be less than 1 n {\displaystyle {\frac ...
Indeed, several classic results in random graph theory show that even networks with no real topological structure exhibit the small-world phenomenon, which mathematically is expressed as the diameter of the network growing with the logarithm of the number of nodes (rather than proportional to the number of nodes, as in the case for a lattice).
The word theory in "the theory of evolution" does not imply scientific doubt regarding its validity; the concepts of theory and hypothesis have specific meanings in a scientific context. While theory in colloquial usage may denote a hunch or conjecture, a scientific theory is a set of principles that explains an observable phenomenon in natural ...
Special education (also known as special-needs education, aided education, alternative provision, exceptional student education, special ed., SDC, and SPED) is the practice of educating students in a way that accommodates their individual differences, disabilities, and special needs. This involves the individually planned and systematically ...
Three examples of Turing patterns Six stable states from Turing equations, the last one forms Turing patterns. The Turing pattern is a concept introduced by English mathematician Alan Turing in a 1952 paper titled "The Chemical Basis of Morphogenesis" which describes how patterns in nature, such as stripes and spots, can arise naturally and autonomously from a homogeneous, uniform state.
is the gradient, i.e., rate of change with position, of the logarithm of the salt concentration, which is equivalent to the rate of change of the salt concentration, divided by the salt concentration – it is effectively one over the distance over which the concentration decreases by a factor of e. The above equation is approximate, and ...
In the probability theory field of mathematics, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. [1] [2] It was first proved by the French mathematician Michel Talagrand. [3] The inequality is one of the manifestations of the concentration of measure phenomenon. [2]
Developmental systems theory, by contrast, assumes that the process/data distinction is at best misleading and at worst completely false, and that while it may be helpful for very specific pragmatic or theoretical reasons to treat a structure now as a process and now as a datum, there is always a risk (to which reductionists routinely succumb ...