Some relations between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$

Abstract

Let Z and N be the set of integers and the set of positive integers, respectively. For a, b, c, d, n ∈ N let N (a, b, c, d; n) be the number of representations of n by ax 2 + by 2 + cz 2 + dw 2 , and let t(a, b, c, d; n) be the number of representations of n by ax(x − 1)/2 + by(y − 1)/2 + cz(z − 1)/2 + dw(w − 1)/2 (x, y, z, w ∈ Z). In this paper we reveal some connections between t(a, b, c, d; n) and N (a, b, c, d; n).

Gap‐Mode Plasmon‐Induced Photovoltaic Effect in a Vertical Multilayer Graphene Homojunction

“…Unfortunately, the photoresponse time could not be measured, but it is expected to be shorter than a few tens of picoseconds, much beyond the photoexcited carrier relaxation time of graphene. [26] In addition, our vertically stacked structure allows fast separation and collection of photoexcited carriers because of the short channel length corresponding to the thickness of MLG (≈2.4 nm). [17] This is also significantly shorter than the carrier diffusion length of graphene (≈15-70 nm).…”

A novel lattice traffic flow model on a curved road

“…Soon thereafter, Nagatani presented some improved lattice models including triangular lattice model, 22 two-dimensional lattice model, 23 highdimensional lattice model 24 and analyzed jamming transition in these models. [20][21][22][23][24][25] Since then, a number of new lattice models 10,[26][27][28][29][30][31][32][33][34][35][36][37][38][40][41][42][43][44][45][46] considering di®erent factors were successively proposed in order to gain a good understanding of complex behavior of the vehicular tra±c°ow.…”

“…From model structure aspect, the relative current was taken into account in the lattice hydrodynamics models, 27,34,36 or the optimal tra±c current is considered. 26,28,29,37,41 Furthermore, two-lane density di®erence lattice was presented.…”

Multiple optimal current difference effect in the lattice traffic flow model

“…Of this, some was extended by position (density) information, 35,36 and others considered the relative current information. 37,38 As well as, the factors both position and relative current information have been simultaneously taken into account in a few lattice models. 39 It is easy to prove that these extended lattice models mentioned above can enhance the stabilization of the traffic system in some degree.…”