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In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.
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mathematics and theorem
In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.
Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.
In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.
In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.
* Crystallographic restriction theorem, in mathematics
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.
The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.
Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development.
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.
In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.
Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.
The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics.
In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.
In mathematics, the Hahn – Banach theorem is a central tool in functional analysis.
Of course, our understanding of what the theorem really means gains in profundity as the mathematics around the theorem grows.
In mathematics, the Poincaré conjecture ( ; ) is a theorem about the characterization of the three-dimensional sphere ( 3-sphere ), which is the hypersphere that bounds the unit ball in four-dimensional space.
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.
On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.
Some, on the other hand, may be called " deep ": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order ( number of elements ) of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange.
mathematics and is
This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.
Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.
But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.
Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.
In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.
So, too, is the mathematical competence of a college graduate who has majored in mathematics.
The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.
The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.
The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.
In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.
There is no prize awarded for mathematics, but see Abel Prize.
mathematics and statement
A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: " Read Euler, read Euler, he is the master of us all.
In mathematics, a lemma ( plural lemmata or lemmas ) from the Greek λῆμμα ( lemma, “ anything which is received, such as a gift, profit, or a bribe ”) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.
One statement of this philosophy is the thesis that mathematics is not created but discovered.
A lucid statement of this is found in an essay written by the British mathematician G. H. Hardy in defense of pure mathematics.
* Mathematical statement, a statement in logic and mathematics
Although most mathematicians and physicists ( and many philosophers ) would accept the statement " mathematics is a language ", linguists believe that the implications of such a statement must be considered.
This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
In mathematics, the Cauchy integral theorem ( also known as the Cauchy – Goursat theorem ) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
In mathematics, a proof is a demonstration that if some fundamental statements ( axioms ) are assumed to be true, then some mathematical statement is necessarily true.
In mathematics, an inequation is a statement that an inequality holds between two values.
In mathematics, the proof of a " some " statement may be achieved either by a constructive proof, which exhibits an object satisfying the " some " statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one.
Famously, he stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any " ignorabimus " ( statement that the truth can never be known ).
That is, if a professor of mathematics makes a statement about numbers, it will be assumed to be true in the absence of evidence to the contrary.
Assuming something to be true for all numbers when it has been shown for over 906 million cases would not generally be considered hasty, but in mathematics a statement remains a conjecture until it is shown to be universally true.
* Difference ( mathematics ), a statement about the relative size or order of two objects
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions.
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