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n vectors X_1, X_2, ..., X_n are linearly dependent iff there exist scalars c_1, c_2, ..., c_n, not all zero, such that sum_(i=1)^nc_iX_i=0. (1) If no such scalars exist, ...

In particle physics, a spinor field of order 2s describes a particle of spin s, where s is an integer or half-integer. Therefore, a spinor of order 4s contains as much ...

A divergenceless field can be partitioned into a toroidal and a poloidal part. This separation is important in geo- and heliophysics, and in particular in dynamo theory and ...

Let ||f|| be the supremum of |f(x)|, a real-valued function f defined on (0,infty). If f is twice differentiable and both f and f^('') are bounded, Landau (1913) showed that ...

In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number h(-d) of an imaginary quadratic field with binary quadratic form discriminant ...

Let lambda be (possibly complex) eigenvalues of a set of random n×n real matrices with entries independent and taken from a standard normal distribution. Then as n->infty, ...

The notion of a Hilbert C^*-module is a generalization of the notion of a Hilbert space. The first use of such objects was made by Kaplansky (1953). The research on Hilbert ...

The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common ...

The constant s_0 in Schnirelmann's theorem such that every integer >1 is a sum of at most s_0 primes. Of course, by Vinogradov's theorem, it is known that 4 primes suffice ...

The nth root (or "nth radical") of a quantity z is a value r such that z=r^n, and therefore is the inverse function to the taking of a power. The nth root is denoted ...