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Bifurcation theory has been applied to connect quantum systems to the dynamics of their classical analogues in atomic systems, [6] [7] [8] molecular systems, [9] and resonant tunneling diodes. [10] Bifurcation theory has also been applied to the study of laser dynamics [ 11 ] and a number of theoretical examples which are difficult to access ...
Hasse–Arf theorem (local class field theory) Hilbert's theorem 90 (number theory) Isomorphism extension theorem (abstract algebra) Joubert's theorem ; Lagrange's theorem (number theory) Mason–Stothers theorem (polynomials) Polynomial remainder theorem (polynomials) Primitive element theorem (field theory) Rational root theorem (algebra ...
Contributing structures of the carbonate ion. In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures (or forms, [1] also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory.
In mathematics and electronics, cavity perturbation theory describes methods for derivation of perturbation formulae for performance changes of a cavity resonator. These performance changes are assumed to be caused by either introduction of a small foreign object into the cavity, or a small deformation of its boundary.
In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus . [ 1 ]
Theorem — The number of strictly positive roots (counting multiplicity) of is equal to the number of sign changes in the coefficients of , minus a nonnegative even number. If b 0 > 0 {\displaystyle b_{0}>0} , then we can divide the polynomial by x b 0 {\displaystyle x^{b_{0}}} , which would not change its number of strictly positive roots.
Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
Solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency , i.e., the beam can resonate. The natural frequencies of a beam therefore correspond to the frequencies at which resonance can occur.