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Diamagnetic Inequality
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... and physics, the diamagnetic inequality relates the Sobolev space, Sobolev norm of the absolute value of a section (fiber bundle), section of a line bundle to its covariant derivative. The diamagnetic inequality has an important physical interpretation, that a electric charge, charged particle in a magnetic field has more energy in its ground state than it would in a vacuum. To precisely state the inequality, let L^2(\mathbb R^n) denote the usual Hilbert space of square-integrable functions, and H^1(\mathbb R^n) the Sobolev space of square-integrable functions with square-integrable derivatives. Let f, A_1, \dots, A_n be measurable functions on \mathbb R^n and suppose that A_j \in L^2_ (\mathbb R^n) is real-valued, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
