While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content.

While in normal typography these are just alternate font variants, they may have different meanings as mathematical symbols.

While some of Euler's proofs are not acceptable by modern standards of mathematical rigour ( in particular his reliance on the principle of the generality of algebra ), his ideas led to many great advances.

While the patent was initially rejected by the patent office as being a purely mathematical invention, following 12 years of appeals, Pardo and Landau won a landmark court case at the CCPA ( Predecessor Court of the Federal Circuit ) overturning the Patent Office in 1983 — establishing that " something does not cease to become patentable merely because the point of novelty is in an algorithm.

While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of the area requires a lot of care.

While other such approaches have been developed, those based on moments are advantageous in terms of mathematical and computational simplicity.

While these result in some contradictions and lack completeness in a mathematical sense ( being based on axioms that are " merely " self-evident, but not robust under the stresses of radical scepticism ), the paradoxes can nonetheless be analysed rationally using the subatomic applications of quantum cosmology-particularly through the employment of the Schrödinger wave equations.

While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean ; this is a task for the physical sciences.

While it is often assumed that those with this intelligence naturally excel in mathematics, chess, computer programming and other logical or numerical activities, a more accurate definition places less emphasis on traditional mathematical ability and more on reasoning capabilities, recognizing abstract patterns, scientific thinking and investigation and the ability to perform complex calculations.

While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 ( and possibly even earlier ) his work remained unknown to most of the mathematical community until years later,

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

While the advanced mathematical nature of such adaptive systems has kept neural networks for financial analysis mostly within academic research circles, in recent years more user friendly neural network software has made the technology more accessible to traders.

While the complexity of real food webs connections are difficult to decipher, ecologists have found mathematical models on networks an invaluable tool for gaining insight into the structure, stability, and laws of food web behaviours relative to observable outcomes.

While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether ( or in what way ) these theorems answer Hilbert's second problem.

While infinitesimals eventually disappeared from the calculus, their mathematical study continued through the work of Levi-Civita and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich ( 2006 ).

While they have useful mathematical properties, they can produce counter-intuitive results: an event with an 80 % probability of occurring is four times more likely to happen than an event with a 20 % probability, but the odds are 16 times higher on the less likely event ( 4 – 1 against, or 4 ) than on the more likely one ( 1 – 4, or 4 – 1 on, or 0. 25 ).

While some coders specialize in developing system-level functionality ( such as providing wrappers and APIs for other coders to base their code on ), others code effects which are usually visual representations of mathematical formulas, such as fractals or metaballs.

" While Javits said any government official who sought to use the bill for busing purposes " would be making a fool of himself ," two years later the Department of Health, Education and Welfare said that Southern school districts would be required to meet mathematical ratios of students by busing.

While admitting that it made him " a little sore ", Patrick has said that the research he and his students conducted was seminal in developing mathematical models against which further research could be compared.

While complex problems can be broken down into simpler multiplications, the answers to these basic operations themselves are essential for more complex mathematical operations.

While content validity depends on a theoretical basis for assuming if a test is assessing all domains of a certain criterion ( e. g. does assessing addition skills yield in a good measure for mathematical skills?

While this was a little outside the main centres of British mathematical research, Burnside remained a very active researcher, publishing more than 150 papers in his career.

While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantified and described.

While the Trivers – Willard hypothesis predicts that birth sex is dependent on living conditions ( i. e. more male children are born in " good " living conditions, while more female children are born in poorer living conditions ), the probability of having a child of either gender is still regarded as 50 / 50.

While the Kolmogorov – Smirnov test is usually used to test whether a given F ( x ) is the underlying probability distribution of F < sub > n </ sub >( x ), the procedure may be inverted to give confidence limits on F ( x ) itself.

While there is a real theorem that a random variable will reflect its underlying probability over a very large sample, the law of averages typically assumes that unnatural short-term " balance " must occur.

While it is possible to define some arbitrary, ad hoc cost function, frequently a particular cost will be used, either because it has desirable properties ( such as convexity ) or because it arises naturally from a particular formulation of the problem ( e. g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost ).

While uranium-238 has a small probability for spontaneous fission or even induced fission with fast neutrons, uranium-235 and to a lesser degree uranium-233 have a much higher fission cross-section for slow neutrons.

Barlow and Proschan define availability of a repairable system as " the probability that the system is operating at a specified time t ." While Blanchard gives a qualitative definition of availability as " a measure of the degree of a system which is in the operable and committable state at the start of mission when the mission is called for at an unknown random point in time.

While the probability of encountering food, potential sexual partners, or predators is the same in all directions for animals floating freely in the water, there is a clear difference on the seafloor between the sides facing toward and away from the substrate, and between their orientation and the vertical direction perpendicular to the substrate.

While both fuzzy logic and probability theory can be used to represent subjective belief, fuzzy set theory uses the concept of fuzzy set membership ( i. e., how much a variable is in a set ), and probability theory uses the concept of subjective probability ( i. e., how probable do I think that a variable is in a set ).

While this distinction is mostly philosophical, the fuzzy-logic-derived possibility measure is inherently different from the probability measure, hence they are not directly equivalent.

While measuring the spin of these entangled particles along parallel axes will always result in opposite ( i. e., perfectly anticorrelated ) results, measurement at perpendicular directions will have a 50 % chance of matching ( i. e., will have a 50 % probability of an uncorrelated result ).

While this reasoning error often can produce a relatively good approximation of the correct hypothesis probability value, it can lead to a completely wrong result and wrong conclusion in case the base rate is very low and the reliability of the test is not perfect.

While anger can activate aggression or increase its probability or intensity, it is neither a necessary nor a sufficient condition for aggression.

While he interpreted the beginning of the year at the vernal equinox, Arthur Anthony Macdonell 1917 stated that the twelve intercalary days " in all probability " were inserted at the winter solstice.

While there, Rabin invented the Miller-Rabin primality test, a randomized algorithm that can determine very quickly ( but with a tiny probability of error ) whether a number is prime.

While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease ; the fluctuation theorem precisely quantifies this probability.

While there is no difference between almost surely and surely ( that is, entirely certain to happen ) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of infinity.

While arguably not philosophically superior to permissible or allowable stress design, it does have the potential to produce a more consistently designed structure as each element is intended to have the same probability of failure.

While the availability heuristic is an effective strategy in many situations, when judging probability the availability heuristic can lead to systematic errors.

While in the program, students gain experiences designed to heighten not only an interest in public service but also their probability of entering into and succeeding in a career in civic leadership.