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We prove that there exists a constant 0<α0≪1 independent of σ>0 such that for all |→p|<1/3 and all values of the finestructure constant 0<α<α0, there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum.
There exists a ground state solution v of equation (1.1).
Hence, there exists a ground state solution of boundary value problem (5.1).
(i) There exists a ground state solution v of equation (1.1).
When a line of sight exists, a ground reflection is added if the distance between the transmitter and the receiver satisfies the following equation: (3).
Assume the hypotheses (H1 - H5) hold, boundary value problem (1.1) has a weak solution such that (I u)=m), i.e. there exists a ground state solution of boundary value problem (1.1).
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Beyond the bright lights and tourist attractions exists a breeding ground for the savvy and assertive, driven by a relentless work ethic and spirit of achievement that draws people from across the globe.
Under assumptions ( f 1 ) ∼ ( f 3 ) and ( V 1 ), there exists a least-energy ground state solution of (1.1).
Under assumptions ( f 1 ) ∼ ( f 4 ) and ( V 2 ), there exists a least-energy ground state of (1.1), which is radially uniqueness solution, with the corresponding least-energy value c ∗.
As the Dagoty brothers expanded the premises, the main building came to resemble what exists today: a ground floor, two upper floors, and an attic lit by seven casement windows that look out onto the street and seventeen others overlooking a cobblestoned courtyard.
If evolutionary theory can demonstrate that religion is beneficial over all, then there might exist a common ground with true believers in religion, who from a very different perspective also argue that the benefits of belief outweigh the costs.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com