We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

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We propose an operator Hermite polynomial method, namely, to replace the special functions' argument by quantum mechanical operator, and in this way we have derived two binomial theorems related to two-variable Hermite polynomials. This method is concise and may be of help in deducing many operator identities, which may become a new branch in mathematical physics theory.

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