These results are new since, to the best of our knowledge, no multiplicity results for gradient systems have appeared in the literature for the case that z = 0 is a saddle point of Φ.
It resists privileging any single discipline of knowledge, preserving a multiplicity that implicates truth in the problem of aesthetic representation.
In the absence of prior knowledge on the multiplicity of pairings between the two protein sets, it is necessary that the protein representatives be unique for each species.
Knowledge of the multiplicity of HIV-1 transmission is a critical component of successful public health management of the HIV/AIDS epidemic, and advancement of HIV-1 transmission prevention.
Indigenous peoples of the Pamirs have maintained knowledge of a multiplicity of medicinal plants through periods of socio-cultural and ecological change.
That common ground, she thinks, is only increased by technological advances, and is prompting philosophers to ask sharp questions about whether the two fields are ultimately distinct, or deeply compatible and therefore good evidence for the "multiplicity of knowledge".
To the best of our knowledge, the existence and multiplicity of solutions to problem (1.1) has not ever been studied by variational methods.
To the best of our knowledge, the existence and multiplicity of soliton solutions to problem (1.1) has never been studied by variational methods.
However, to the best of our knowledge, the existence and multiplicity of positive solutions for (1.1) are relatively few by the fixed point index theory.
To the best of our knowledge, the existence and multiplicity of nontrivial solutions to system (1.1) has never been studied by variational methods, where g is a more general function, f ( x, u ) ≠ | u | p − 1 u is also general, and V ( x ) and K ( x ) may be non-radial symmetrical and non-periodic.
