By mathematical arguments, the maximum must lie between the two minima.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them ; we are studying the relationships between various classes of mathematical structures.

* By adding bits to each encoded unit, the redundancy allows both to detect errors in coded data and to correct them based on mathematical algorithms.

* By adding bits to each encoded unit, the redundancy allows both to detect errors in coded data and to correct them based on mathematical algorithms.

By simply rewriting the mathematical expression one can save 50 % of the multiplications required.

By 1901, Peano was at the peak of his mathematical career.

By revealing ( in modern terms ) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history ; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.

By means of mathematical induction, it is easily proven that the above procedure requires the minimal number of moves possible, and that the produced solution is the only one with this minimal number of moves.

By the principle of mathematical induction it follows that the result is true for all natural numbers.

By 1928, Gamow had solved the theory of the alpha decay of a nucleus via tunnelling, with mathematical help from Nikolai Kochin.

By the time of the Surrealist Exhibition of Objects in 1936 a whole range of sub-classifications had been devised — including found objects, readymade objects, perturbed objects, mathematical objects, natural objects, interpreted natural objects, incorporated natural objects, Oceanic objects, American objects and Surrealist objects.

By using counterexamples to show that certain conjectures are false, mathematical researchers avoid going down blind alleys and learn how to modify conjectures to produce provable theorems.

By this Polykleitos meant that a statue should be composed of clearly definable parts, all related to one another through a system of ideal mathematical proportions and balance, no doubt expressed in terms of the ratios established by Pythagoras for the perfect intervals of the musical scale: 1: 2 ( octave ), 2: 3 ( harmonic fifth ), and 3: 4 ( harmonic fourth ).

By the 13th century, Hindu-Arabic numerals were accepted in European mathematical circles ( Fibonacci used them in his Liber Abaci ).

By this change, it will gain in utility, interest and real truth far more than a full compensation for the forfeiture of a fictitious title to mathematical exactness and certainty.

By using eccentrics and epicycles, his geometrical model achieved greater mathematical detail and predictive accuracy than had been exhibited by earlier concentric spherical models of the cosmos.

By assigning " energy values " to the various psychological components of information metabolism, Augustinavičiūtė created a mathematical theory of thinking.

By making simultaneous measurements at the two different frequencies, the resulting data enabled theoretical physicists to verify the mathematical predictions of Albert Einstein's General Theory of Relativity.

By calculating the area of the peak using the mathematical function of integration, the concentration of an analyte in the original sample can be determined.

By staying away from complete mathematical rigor while emphasizing the physical and engineering interpretations of probability, Papoulis's book gained wide popularity.

By the early 1990s, when Second Edition versions of Numerical Recipes ( with code in C, Fortran-77, and Fortran-90 ) were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between the more mathematical numerical analysts and the larger community using integrated environments.

: Proof: By mathematical induction.

By undermining the traditional role of art and artist, its humor is reflective of a goal to bring life back into art, which Maciunas states in his agenda, “ If man could experience the world, the concrete world surrounding him ( from mathematical ideas to physical matter ) in the same way he experiences art, there would be no need for art, artists and similar ‘ nonproductive ’ elements ”.

By analogy, the church also has been regarded as entirely independent of the `` world '' in the sense of requiring nothing from it in order to be the church.

By analogy, the phenomenon of small events causing similar events leading to eventual catastrophe is called the domino effect.

By analogy with the word " conlang ", the term conworld is used to describe these fictional worlds, inhabited by fictional constructed cultures.

By analogy, the name hammer has also been used for devices that are designed to deliver blows, e. g. in the caplock mechanism of firearms.

By analogy with the ancient Greek term for agriculture, geoponics, the science of cultivating the earth, Gericke coined the term hydroponics in 1937 ( although he asserts that the term was suggested by W. A. Setchell, of the University of California ) for the culture of plants in water ( from the Greek hydro -, " water ", and ponos, " labour ").

By analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system, in the form

By analogy, stone arches are irreducibly complex — if you remove any stone the arch will collapse — yet we build them easily enough, one stone at a time, by building over centering that is removed afterward.

By analogy, the term letter is sometimes used for e-mail messages with a formal letter-like format.

By analogy a typescript has been produced on a typewriter.

By analogy with the phoneme, linguists have proposed other sorts of underlying objects, giving them names with the suffix-eme, such as morpheme and grapheme.

By analogy the same term is used in politics and public affairs to refer to the informal process by which statements, designed to refute or negate specific arguments put forward by opponents, are deployed in the media.

By analogy, the word " signature " may be used to refer to the characteristic expression of a process or thing.

By analogy, a similar graph depicting the progress of a string as time passes by can be obtained ; the string ( a one-dimensional object — a small line — by itself ) will trace out a surface ( a two-dimensional manifold ), known as the worldsheet.

" In 1994, the Oxford Dictionary of Philosophy offered a wider definition: " By analogy with racism and sexism, the improper stance of refusing respect to the lives, dignity, or needs of animals of other than the human species.

By explaining past changes by analogy with present phenomena, a limit is set to conjecture, for there is only one way in which two things are equal, but there are an infinity of ways in which they could be supposed different.

By analogy, principles in other fields of study are sometimes loosely referred to as " laws ".

By analogy, the term " silver aten " was sometimes used to refer to the moon.

By this analogy, it is suggested, the experience of free will emerges from the interaction of finite rules and deterministic parameters that generate nearly infinite and practically unpredictable behaviourial responses.

By analogy, this episode of the myth was eventually equated with other interactions between a human and a being in the divine realm.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By way of analogy, the allele ( a particular version of a gene ) which causes sickle-cell anemia when two copies are present may also confer resistance to malaria with a lesser form of anemia when one copy is present ( this is called heterozygous advantage ).

By analogy to the neurotic behavior in monkeys, he suggested that these developmental abnormalities are a major cause of adult violence amongst humans.

By analogy, the magnetic equation is an inductive current involving spin.

By analogy with monozygotic and dizygotic twins, such a combination is called dizygotic triplets.