This textbook also derives generalizations of Itō's lemma for non-Wiener ( non-Gaussian ) processes.

There is no prima facie reason for this substitution error and there is no erroneous parallel to be found with the word lemma, from which dilemma derives.

In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.

The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim-Skolem theorem, lets us sharply reduce the complexity of the generic formula for which we need to prove the theorem:

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.

According to his lemma, a group of four manuscripts including Codex Monacensis 1086 are copies directly from the original.

The morphology functions of the software distributed with the database try to deduce the lemma or root form of a word from the user's input ; only the root form is stored in the database unless it has irregular inflected forms.

Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set.

There is a contravariant version of Yoneda's lemma which concerns contravariant functors from C to Set.

Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that

An important special case of Yoneda's lemma is when the functor F from C to Set is another hom-functor h < sup > B </ sup >.

Both follow easily from the second Borel – Cantelli lemma.

The natural transformations from a representable functor to an arbitrary functor are completely known and easy to describe ; this is the content of the Yoneda lemma.

This follows from the naturality of the sequence produced by the snake lemma.

It follows immediately from the five lemma.

The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B ′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A ′ of B ′ and also an isomorphism from the factor object B / A to B ′/ A ′', then f itself is an isomorphism.

This follows from Schur's lemma.

The maps δ < sup > n </ sup > are called the " connecting homomorphisms " and can be obtained from the snake lemma.

But the general case follows from the projective case with the aid of Chow's lemma.

Sperner's lemma, from 1928, states that every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors.

Both of these naturalities follow from the naturality of the sequence provided by the snake lemma.

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it.

Instead, we give a sketch of how one can derive Itō's lemma by expanding a Taylor series and applying the rules of stochastic calculus.

One of these, Itō's lemma, expresses the composite of an Itō process ( or more generally a semimartingale ) dX < sub > t </ sub > with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dX < sub > t </ sub > and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way.

around f, then by the fundamental lemma of calculus of variations.

Using Itō's lemma with f ( S )

Applying Itō's lemma with f ( S ) =

Applying Itō's lemma with f ( Y ) = log ( Y ) gives

Then, if the value of an option at time t is f ( t, S < sub > t </ sub >), Itō's lemma gives

To prove Fitting's lemma, we take an endomorphism f of M and consider the following two sequences of submodules.

By Schur's lemma, the image of f ″ is a G × G irreducible representation, which is therefore n × n dimensional, which also happens to be a subrepresentation of C ( f ″ is nonzero ).

where e is the identity of G. Though the f ″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G × G-irreducible subrepresentation of C, we have just showed that as an irreducible G × G-subrepresentation of C is contained in A ⊗ B as another irreducible G × G-subrepresentation of C. Using Schur's lemma again, this means both irreducible representations are the same.

Sard's theorem, also known as Sard's lemma or the Morse – Sard theorem, is a result in mathematical analysis which asserts that the image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set.

We first have to show that the ascending Kleene chain of f exists in L. To show that, we prove the following lemma:

* If f is L < sup > 1 </ sup > integrable and supported on ( 0, ∞), then the Riemann – Lebesgue lemma also holds for the Laplace transform of f. That is,

: This follows by extending f by zero outside the interval, and then applying the version of the lemma on the entire real line.

Using the lemma, one can " lift " a root r of the polynomial f mod p < sup > k </ sup > to a new root s mod p < sup > k + 1 </ sup > ( by taking m =

By working directly in the p-adics and using the p-adic absolute value, there is a version of Hensel's lemma which can be applied even if we start with a solution of f ( a ) ≡ 0 mod p such that f '( a ) ≡ 0 mod p. We just need to make sure the number f '( a ) is not exactly 0.

" Urdi's lemma " was an extension of Apollonius ' theorem that allowed an equant in an astronomic model to be replaced with an equivalent epicycle that moved around a deferent centered at half the distance to the equant point.