A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

Fundamentals of Diatomic Molecular Spectroscopy

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of heuristic nature and are actually not needed in analysis of diatomic spectra. This work addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarises aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

The Floating of a Body on the Surface of a Liquid: A Synthesis

The current physical interpretation of the phenomenon of floating of a body on the surface of a liquid - the so-called ’Archimedes’ Principle’ - is not correct. It is based on a assumption which is false, because it leads to contradict observed facts. A different physical interpretation is the correct interpretation, which is equivalent to the current one from the quantitative viewpoint, but significantly different from the physical viewpoint. When a volume of matter – a body - is immersed in a fluid, it 'loses' its own weight, i.e., it is no longer home to the downward-directed mechanical action of which the volume is home when it is immersed in a vacuum. Therefore, a material volume immersed in a fluid does not host two mechanical actions - the 'weight' of the volume itself, directed downwards, and the 'Archimedes' force', directed upwards – but a single mechanical action. The intensity of this action is proportional to the volume of the body and to the difference in density between the matter which constitutes the body and the matter which constitutes the fluid. The direction of this action is related to the sign of the difference between the two values of density, reversing as the sign changes from positive to negative and vice-versa. Since the intensity of this mechanical action is proportional to the difference in density between the matter which constitutes the body and the matter which constitutes the fluid, there is no mechanical action – i.e., the mechanical action ceases – if there is equilibrium in density between the matter of the body and the matter of the fluid. This indicates that the main cause of this mechanical action is a physical disequilibrium between the matter which constitutes the body and the matter which constitutes the fluid, i.e., a physical disequilibrium/relationship between matter which is in contact.

Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ - Δ u - k 2 u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p R N with $$k>0,$$ k > 0 , $$N \ge 3$$ N ≥ 3 , $$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$ p ∈ 2 ( N + 1 ) N - 1 , 2 N N - 2 and $$Q \in L^{\infty }({\mathbb {R}}^{N})$$ Q ∈ L ∞ ( R N ) . Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

Plahte diagrams for string scattering amplitudes

Plahte identities are monodromy relations between open string scattering amplitudes at tree level derived from the Koba-Nielsen formula. We represent these identities by polygons in the complex plane. These diagrams make manifest the appearance of sign changes and singularities in the analytic continuation of amplitudes. They provide a geometric expression of the KLT relations between closed and open string amplitudes. We also connect the diagrams to the BCFW on-shell recursion relations and generalise them to complex momenta resulting in a relation between the complex phases of partial amplitudes.