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In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It shows that the analog of the Cauchy–Kovalevskaya theorem does not hold in the smooth category.
When the topic is a theorem, the article should provide a precise statement of the theorem. Sometimes this statement will be in the lead, for example: 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.
It is inspired by the typographic practice of end marks, an element that marks the end of an article. [1] [2] In Unicode, it is represented as character U+220E ∎ END OF PROOF. Its graphic form varies, as it may be a hollow or filled rectangle or square. In AMS-LaTeX, the symbol is automatically appended at the end of a proof environment ...
Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed out in 1859 one of the lemmas used by Legendre is false. Dirichlet gave a complete proof in 1837. The proofs of the Kronecker–Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first ...
Fuchs's theorem (differential equations) Fuglede's theorem (functional analysis) Full employment theorem (theoretical computer science) Fulton–Hansen connectedness theorem (algebraic geometry) Fundamental theorem of algebra (complex analysis) Fundamental theorem of arbitrage-free pricing (financial mathematics)
This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form. It is not the most general form. Let V {\displaystyle V} be a real Hilbert space and a ( ⋅ , ⋅ ) {\displaystyle a(\cdot ,\cdot )} a bilinear form on V {\displaystyle V} , which is
In mathematics, a Thue equation is a Diophantine equation of the form f ( x , y ) = r , {\displaystyle f(x,y)=r,} where f {\displaystyle f} is an irreducible bivariate form of degree at least 3 over the rational numbers , and r {\displaystyle r} is a nonzero rational number.
To simplify the notation, let = ˙ and define a collection of n 2 functions Φ j i by =. Theorem. (Douglas 1941) There exists a Lagrangian L : [0, T] × TM → R such that the equations (E) are its Euler–Lagrange equations if and only if there exists a non-singular symmetric matrix g with entries g ij depending on both u and v satisfying the following three Helmholtz conditions: