Proof: By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s

By 1990, Banks went on to produce Spice 1's 187 Proof E. P., another successful independent project that he put his signature production on, selling more than 200, 000 units.

* Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects

Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

* Metamath Proof explorer a repository of mathematical theorems along with their proofs using the Metamath proof assistant.

* Proof by exhaustion or brute force method, a method of mathematical proof

Proof by exhaustion, also known as proof by cases, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases and each case is checked to see if the proposition in question holds.

Proof that QCD confines at low energy is a mathematical problem of great relevance, and an award has been proposed by the Clay Mathematics Institute for whoever is also able to show that the Yang – Mills theory has a mass gap and its existence.

: Proof theory is the mathematical study of formalized arguments.

Newman also wrote Gödel's Proof ( 1958 ) with Ernest Nagel, presenting the main results of Gödel's incompleteness theorem and the mathematical work and philosophies leading up to its discovery in a more accessible manner.

This book inspired Douglas Hofstadter to take up the study of mathematical logic, write his famous book Gödel, Escher, Bach, and prepare a second edition of Gödel's Proof, published in 2002.

: Proof: We use mathematical induction.

Proof by induction:

** Proof by induction

Proof.

Proof.

Proof.

Proof.

Proof.

* Sanjeev Saxena, " A Simple Proof of Bernoulli's Inequality ", viXra: 1205. 0068, May 2012

Generally this is in a very small detail, such as the number of leaves on the ear of corn on the recent US Wisconsin state quarter: File: 2004 WI Proof. png.

: Proof: Let A be infinite RE.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

‡ Proof.

Proof.

Proof.

Proof.

Proof.

Proof: Suppose that and are two identity elements of.

Proof: Suppose that and are two inverses of an element of.

Proof.

Proof.

Proof.

Proof.

A more successful effort was the Standard Proof for Pythagoras ' Theorem, that replaced the more than 100 incompatible existing proofs.

* Demonstratio evangelica ( Proof of the Gospel ) is closely connected to the Praeparatio and comprised originally twenty books of which ten have been completely preserved as well as a fragment of the fifteenth.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.

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 mathematical analogy: A metasyntactic variable is a word that is a variable for other words, just as in algebra letters are used as variables for numbers.

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.

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 ”.