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This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
where = is the reduced Planck constant.. The quintessentially quantum mechanical uncertainty principle comes in many forms other than position–momentum. The energy–time relationship is widely used to relate quantum state lifetime to measured energy widths but its formal derivation is fraught with confusing issues about the nature of time.
Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation ...
List of mathematical functions; List of mathematical identities; List of mathematical proofs; List of misnamed theorems; List of scientific laws; List of theories; Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.
This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form. It is not the most general form. Let V {\displaystyle V} be a real Hilbert space and a ( ⋅ , ⋅ ) {\displaystyle a(\cdot ,\cdot )} a bilinear form on V {\displaystyle V} , which is
Monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no satisfying () =: this happens, for example, when is a Dirac measure but is not. We can improve on this by adopting Kantorovich's formulation of the optimal transportation problem, which is to find a probability measure γ {\displaystyle ...
"The linear complementarity problem, sufficient matrices, and the criss-cross method" (PDF). Linear Algebra and Its Applications. 187: 1– 14. doi: 10.1016/0024-3795(93)90124-7. Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones" (PDF).