相关论文: Thirty-Six Unsolved Problems in Number Theory
Quantum gravity is the missing piece in our understanding of the fundamental interactions today. Given recent observational breakthroughs in gravity, providing a quantum theory for what lies beyond general relativity is more urgent than…
This paper concerns the number of lattice points in a circle.
Since its discovery, quantum theory has proven to be one of the most precise theories ever made. Measurement processes, however, do not seem to be governed by the unitary law of quantum mechanics, and one can ask if the theory is complete.…
In this note we briefly survey and propose some open problems related to isoparametric theory.
We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three…
Black holes in several dimensions and in several theories are studied and discussed. The theories are, general relativity, Kaluza-Klein, Brans-Dicke, Lovelock gravity and string theory.
Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known…
Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of…
The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic…
The role of geometrically infinitely divisible laws in renewal equations and superposition of renewal processes are explored here. Some examples are also discussed.
In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if…
The theory of quadratic forms and class numbers has connections to many classical problems in number theory. Recently, class numbers have appeared in the study of black holes in string theory. We describe this connection and raise questions…
The universe we see gives every sign of being composed of matter. This is considered a major unsolved problem in theoretical physics. Using the mathematical modeling based on the algebra ${\bf{T}} := {\bf{C}}\otimes{\bf{H}}\otimes{\bf{O}}$,…
This is a survey of old and new problems and results in additive number theory.
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
The notion of geometric version of an infinitely divisible law is introduced. Concepts parallel to attraction and partial attraction are developed and studied in the setup of geometric summing of random variables.
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
In these lectures I discuss various unsolved problems of string theory and their relations to quantum gravity, 3d Ising model, large N QCD, and quantum cosmology. No solutions are presented but some new and perhaps useful approaches are…
The concepts of symmetry and its breakdown are investigated in two different terms according to whether the resulting asymmetry is universal or only obtained for a special configuration: we shall illustrate this by considering in the first…
We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a…