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For functions in certain classes, the problem of determining: whether two functions are equal, known as the zero-equivalence problem (see Richardson's theorem); [5] the zeroes of a function; whether the indefinite integral of a function is also in the class. [6] Of course, some subclasses of these problems are decidable.
Construct a finite nilpotent loop with no finite basis for its laws. Proposed: by M. R. Vaughan-Lee in the Kourovka Notebook of Unsolved Problems in Group Theory; Comment: There is a finite loop with no finite basis for its laws (Vaughan-Lee, 1979) but it is not nilpotent.
Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O (2 p ( n ) ) time, where p ( n ) is a polynomial function of n .
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
For instance if a nonstandard (non-finite) element u is in the model, then so is m ⋅ u for any m in the initial segment N, yet u 2 is larger than m ⋅ u for any standard finite m. Also one can define "square roots" such as the least v such that v 2 > 2 ⋅ u. These cannot be within a standard finite number of any rational multiple of u.
7th: Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? Resolved. Result: Yes, illustrated by the Gelfond–Schneider theorem. 1934 8th: The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is 1/2") and other prime-number problems, among them Goldbach's conjecture and the twin ...
Every operator on a non-trivial complex finite dimensional vector space has an eigenvector, solving the invariant subspace problem for these spaces. In the field of mathematics known as functional analysis , the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends ...
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1] [2] Nonlinear problems are of interest to engineers, biologists, [3] [4] [5] physicists, [6] [7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature. [8]