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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.
The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
In order to enhance the attractiveness of this book as a textbook, we have included worked-out examples at appropriate points in the text and have included lists of exercises for Chapters 1 — 9. These exercises range from routine problems to alternative proofs of key theorems, but containing also material going beyond what is covered in the text.
There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The notable unsolved problems in statistics are generally of a different flavor; according to John Tukey, [1] "difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." A list of ...
The problem to determine all positive integers such that the concatenation of and in base uses at most distinct characters for and fixed [citation needed] and many other problems in the coding theory are also the unsolved problems in mathematics.
Example problems may well include all of the CMI prize problems. Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove. A method guaranteed to find a proof if a ...
The resolution of the debate has not been agreed upon by economists. Transformation problem : The transformation problem is the problem specific to Marxist economics, and not to economics in general, of finding a general rule by which to transform the values of commodities based on socially necessary labour time into the competitive prices of ...
[4] [14] This has been achieved for several small globular proteins in recent years. [15] In 2020, it was announced that Google's AlphaFold , a neural network based on DeepMind artificial intelligence, is capable of predicting a protein's final shape based solely on its amino-acid chain with an accuracy of around 90% on a test sample of ...