When applied to vector spaces of functions ( which typically are infinite-dimensional ), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces.

When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 days in quarantine before being able to visit Hilbert.

When studying inorganic compounds, one often encounters parts of the different classes of inorganic chemistry ( an organometallic compound is characterized by its coordination chemistry, and may show interesting solid state properties ).

When Beethoven was finished studying with Albrechtsberger he decided to get a few more tips and pointers, so to speak, from Haydn.

When working on a major specimen like an eagle, he would spend up to four 15-hour days, preparing, studying, and drawing it.

When she was not studying, she took independent courses from Ohio University and spent her free time casting bronzes in the school foundry.

When studying the mathematics of games, the mathematical analysis of the game is more important than actually playing the game.

When asked how he developed his mathematical abilities so rapidly, he replied " by studying the masters, not their pupils.

When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics.

When he was eleven, his parents sent him to the Protestant Academy at Sedan, where he spent four years studying Greek under Jacques du Rondel.

When the word " seashells " is used to refer only to the shells of marine mollusks then studying seashells is part of conchology.

When studying the whole molluscan animal is included as well as studying the shell, then the study is known as malacology ; a person who studies mollusks is known as a malacologist.

When this topic turned out not to be feasible, he switched to studying the question of Italian unification over a two-year period.

When the school's high school was founded in the late 1940s, he instituted a number of innovations in the curriculum, including teaching Talmud to boys and girls studying in classes together.

When studying the etiology of a disease, it is better to analyze incidence rather than prevalence since prevalence considers the duration of a condition rather than providing a measure of risk alone.

When Steinberg and Kincheloe wrote the first edition of Kinderculture: The Corporate Construction of Childhood in 1997 ( second edition, 2004 ) many people who made their living studying, teaching, or caring for children were not yet aware of the nature of the changes in childhood that they encountered daily.

When studying the anatomy of the human body, he did not believe that the living organisms could be explained mechanistically.

When Thalberg returned to school, he lacked the stamina to enroll in college which he felt would have required constant late-night studying and cramming for exams.

When he was 17, he moved to the United States with the intention of studying towards eventually entering medical school, but Wang changed his mind and ended up in the arts, studying film and television at California College of Arts and Crafts in Oakland.

When Shunryu had first come to Zoun-in, 8 other boys were studying there.

When about a year old he lost his sight through smallpox ; but this did not prevent him from acquiring a knowledge of Latin and Greek, and studying mathematics.

When Wüthrich joined the Bell Labs, he was put in charge of one of the first superconducting NMR spectrometers, and started studying the structure and dynamics of proteins.

When an axiomatic system begins with its axioms, the primitive notions may not be explicitly stated.

When three dice are tossed repeatedly, what is the probability that the number of aces is 0 ( or 1, or 2, or 3 )??

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ( sometimes termed the atom ’ s “ electron cloud ” ) tends toward a generally spherical zone of probability describing where the atom ’ s electrons will be found.

When the probability of different events is not independent, the probability of future events can change based on the outcome of past events ( see statistical permutation ).

When the probability of repeated events are not known, outcomes may not be equally probable.

When we perform the experiment, the probability of obtaining two particles in the state is now 0. 33 ; the probability of obtaining two particles in the state is 0. 33 ; and the probability of obtaining one particle in the state and the other in the state is 0. 33.

When a coin is fair, that is, the probability of heads is the same as the probability of tails, the entropy of a coin toss is as high as it could be.

When is a conditional probability distribution and the loss function is the negative log likelihood:, then empirical risk minimization is equivalent to maximum likelihood estimation.

When either randomness or uncertainty modeled by probability theory is attributed to such errors, they are " errors " in the sense in which that term is used in statistics ; see errors and residuals in statistics.

When dealing with experiments that are random and well-defined in a purely theoretical setting ( like tossing a fair coin ), probabilities describe the statistical number of outcomes considered divided by the number of all outcomes ( tossing a fair coin twice will yield HH with probability 1 / 4, because the four outcomes HH, HT, TH and TT are possible ).

When it comes to practical application, however, the word probability does not have a singular direct definition.

When we measure this qubit in the standard basis, the probability of outcome is and the probability of outcome is.

When a statistical procedure has been specified in the study protocol, then statistical theory provides well-defined probability statements for the method when applied to all populations that could have arisen from the randomization used to generate the data.

When the stereochemistry of a macromolecule is considered to be a Bernoulli process, triad composition can be calculated from the probability of finding meso diads ( P < sub > m </ sub >).

When this probability is 0. 25 then the probability of finding:

When we consider 100 billion monkeys, the probability falls to 0. 17 %, and as the number of monkeys n increases, the value of X < sub > n </ sub > – the probability of the monkeys failing to reproduce the given text – approaches zero arbitrarily closely.

When thinking about orbitals, we are often given an orbital vision which ( even if it is not spelled out ) is heavily influenced by this Hartree – Fock approximation, which is one way to reduce the complexities of molecular orbital theory.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

When applied to the simplest two-electron molecule, H < sub > 2 </ sub >, valence bond theory, even at the simplest Heitler-London approach, gives a much closer approximation to the bond energy, and it provides a much more accurate representation of the behavior of the electrons as chemical bonds are formed and broken.

When, in the 1970s and 1980s, Jürgen Habermas redefined critical social theory as a theory of communication, i. e. communicative competence and communicative rationality on the one hand, distorted communication on the other, the two versions of critical theory began to overlap or intertwine to a much greater degree than before.

When the intermediate states at intermediate times are energy eigenstates ( collections of particles with a definite momentum ) the series is called old-fashioned perturbation theory.

When describing graviton interactions, the classical theory ( i. e., the tree diagrams ) and semiclassical corrections ( one-loop diagrams ) behave normally, but Feynman diagrams with two ( or more ) loops lead to ultraviolet divergences ; that is, infinite results that cannot be removed because the quantized general relativity is not renormalizable, unlike quantum electrodynamics.

When Charles Darwin proposed his theory of evolution in 1859, one of its major problems was the lack of an underlying mechanism for heredity.

When speaking in terms of a labor theory of value, value, without any qualifying adjective should theoretically refer to the amount of labor necessary to the production of a marketable commodity, including the labor necessary to the development of any real capital employed in the production.

Following World War II, a nationwide movement pressed to return to pre-war society as quickly as possible in the U. S. When combined with the increasing national paranoia about communism and psychoanalytic theory that had become pervasive in medical knowledge, homosexuality became an undesired characteristic of employees working for the U. S. government in 1950.

When Mendel's theories were integrated with the chromosome theory of inheritance by Thomas Hunt Morgan in 1915, they became the core of classical genetics.

When Edward Witten named M-theory, he did not specify what the M stood for — perhaps because the nascent theory was not fully defined.

When k points are blown-up, the del Pezzo surface describes M-theory on a k-torus, and the exceptional del Pezzo surface, namely P < sup > 1 </ sup >( C ) × P < sup > 1 </ sup >( C ), is connected with type IIB string theory in 10 dimensions.

When significant evolutionary change occurs, the theory proposes that it is generally restricted to rare and geologically rapid events of branching speciation called cladogenesis.

When Eldredge and Gould published their 1972 paper, allopatric speciation was considered the " standard " theory of speciation.

When first published, relativity superseded a 200-year-old theory of mechanics stated by Isaac Newton.

When an old theory and a new one are parallel in this way, we can conclude that we are describing the same reality, only more completely.

When not actively performing The Wizard busies himself checking the validity of his cosmology or " theory of everything " which includes elements found in such thinkers as the philosopher Whitehead, the sociologist Parsons, the physicist Prigogine, the systems analyst Jantsch and the biologist Sheldrake.

When a surface is illuminated by a light wave, according to diffraction theory, each point on an illuminated surface acts as a source of secondary spherical waves.

When the technology to perform direct measurements with a pressure probe was developed, there was initially some controversy about whether the classic theory was correct, because some workers were unable to demonstrate negative pressures.

# When theory is based on analogy between explained and unexplained observations, " generally only a limited aspect of the familiar phenomena is incorporated into ( the ) explanatory model ".