The second largest region of phase locking occurs for periods of stimulus that are an integer multiplicity of the intrinsic period.
We found that only when the extrinsic period is close to an integer multiplicity of the averaged intrinsic period can the collective behaviors be induced/enhanced; otherwise, the stimulus possibly ruins the achieved collective behaviors.
On the other hand, in the case of sinusoidal stimulus, the condition under which the external stimulus induces a collective behavior is basically similar to that in the case of periodic impulsive stimulus, that is, only when the extrinsic period is close to an integer multiplicity of the intrinsic period, can the synchronization be achieved.
This is even more true in the case of the asymmetry index (varvec{alpha }(Gamma )) whose precise definition is more technical and requires the use of a certain seminorm defined for integer multiplicity currents [18, Sect. 2].
Let d ∈ ℤ + be a prescribed integer, which corresponds to the maximum multiplicity of chemical graphs, and Σk,denotethethe set of all alternating sequences (c0,…,mc1,…,m k,c k ) consisting of colors c0,c1,…,c k ∈Σ and m1,…,m…,m k ∈{1,2,…,d}.
Over the past decades, the existence and multiplicity of nontrivial solutions for the integer order Schrödinger equation have been extensively investigated.
Suppose that d(≥ 0) is an integer, p z) is an analytic function in D, and the multiplicity of its all zeros is at most d.
Theorem D Suppose that d (≥0) is an integer, p ( z ) is an analytic function in D, and the multiplicity of its all zeros is at most d.
This procedure employs a majority rule, that is, for every label a that appears in at least one of the input trees, the multiplicity of a in ℳ is chosen as the largest integer m such that a appears in at least half of the input trees with multiplicity at least m.
