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Pseudomathematics, or mathematical crankery, is a mathematics-like activity that does not adhere to the framework of rigor of formal mathematical practice. Common areas of pseudomathematics are solutions of problems proved to be unsolvable or recognized as extremely hard by experts, as well as attempts to apply mathematics to non-quantifiable ...
Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R 4 has a negative solution, and in the second of which he proved that it has a positive solution.
In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers (1950, 1951, 1953, 1956) that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.
Pseudo-finite fields and hyper-finite fields are PAC. A non-principal ultraproduct of distinct finite fields is (pseudo-finite and hence [3]) PAC. [2] Ax deduces this from the Riemann hypothesis for curves over finite fields. [1] Infinite algebraic extensions of finite fields are PAC. [4] The PAC Nullstellensatz.
[1] Every element x with the property x* = 0 (or equivalently, x** = 1) is called dense. Every element of the form x ∨ x* is dense. D(L), the set of all the dense elements in L is a filter of L. [1] [2] A distributive p-algebra is Boolean if and only if D(L) = {1}. [1] Pseudocomplemented lattices form a variety; indeed, so do ...
In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F ).
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In mathematics, a pseudogroup is a set of homeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation [dubious – discuss] of the concept of a group, originating however from the geometric approach of Sophus Lie [1] to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example).