In 1995 Kim famously proved the Ramsey bound R(3, t) ≥ ct 2 / log t by constructing an n-vertex graph that is triangle-free and has independence number at most C √ n log n. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph Kn into a packing of such nearly optimal Ramsey R(3, t) graphs. More precisely, for any ǫ > 0 we find an edge-disjoint collection (Gi)i of n-vertex graphs Gi ⊆ Kn such that (a) each Gi is triangle-free and has independence number at most Cǫ √ n log n, and (b) the union of all the Gi contains at least (1 − ǫ) n 2 edges. Our algorithmic proof proceeds by sequentially choosing the graphs Gi via a semi-random (i.e., Rödl nibble type) variation of the triangle-free process.As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabó (concerning a Ramsey-type parameter introduced by Burr, Erdős, Lovász in 1976). Namely, denoting by sr(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H = K3 and establish that sr(K3) = Θ(r 2 log r). The triangle-free process (proposed by Bollobás and Erdős) proceeds as follows: starting with an empty n-vertex graph, in each step a single edge is added, chosen uniformly at random from all non-edges which do not create a triangle.2 Kim's semi-random variation proceeds similar to the triangle-free process, but intuitively adds a large number of carefully chosen random-like edges in each step (instead of just a single edge); see Section 2 for more details. Main result: packing of nearly optimal Ramsey R(3, t) graphsKim and Bohman both proved the Ramsey bound R(3, t) = Ω(t 2 / log t) by showing the existence of a trianglefree graph G ⊆ K n on n vertices with independence number α(G) = O( √ n log n), which is best possible up to the value of the implicit constants. Our first theorem naturally extends their celebrated results, by approximately decomposing the complete graph K n into a packing of such nearly optimal Ramsey R(3, t) graphs.Theorem 1. For any ǫ > 0 there exist n 0 , C, D > 0 such that, for all n ≥ n 0 , there is an edge-disjointOur algorithmic proof proceeds by sequentially choosing the |I| = Θ( n/ log n) edge-disjoint triangle-free subgraphs√ n log n) via a semi-random variation of the triangle-free process akin Kim [20] (see Sections 1.3 and 2 for the details). In particular, we do not only show existence of the (G i ) i∈I , but also obtain a polynomial-time randomized algorithm which constructs these subgraphs. Theorem 1 improves a construction of Fox et.al. [16, Lemma 4.2], who used the basic Lovász Local Lemma based R(3, t)-approach to sequentially choose Θ( √ n/ log n) edge-disjoint triangle-free subgraphs with α(G i ) = O( √ n log n). It is natural to suspect that applying a more sophisticated R(3, t)-approach in each iteration ought to give an improved packing (with smaller independence number than the LLL approach), and here the usage of the triangle-free process was...
In this paper, an electromagnetic variable valve train with a magnetorheological buffer (EMVT with MR buffer) is proposed. This system is mainly composed of an electromagnetic linear actuator (EMLA) and a magnetorheological buffer (MR buffer). The valves of an internal combustion engine are driven by the EMLA directly to open and close, which can adjust the valve lift and phase angle of the engine. At the same time, MR buffer can reduce the seat velocity of the valve and realize the seat buffer of the electromagnetic variable valve. In this paper, the overall design scheme of the system is proposed and the structure design, finite element simulation of the EMLA, and the MR buffer are carried out. The electromagnetic force characteristics of the EMLA and buffer force of the MR buffer are measured, and the seat buffering performance is verified as well. Experiments and simulation results show that the electromagnetic force of the EMLA can reach 320.3 N when the maximum coil current is 40 A. When the current of the buffer coil is 2.5 A and the piston’s motion frequency is 5 Hz, the buffering force can reach 35 N. At the same time, a soft landing can be realized when the valve is seated.
This note contains a refined alteration approach for constructing H-free graphs: we show that removing all edges in H-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdős and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers.
Molecular Dynamics (MD) simulations, supported by parallel software and special hardware, are widely used in materials, computational chemistry and biology science. With advancing of FPGA capability and inclusion of embedded multipliers, lots of studies steer to focus on FPGA accelerated MD simulations. In this paper, we propose a system that can implement the computation on FPGA for Lennard-Jones (LJ) force which has been proved of dominating the whole execution time, and then the results are transferred to the host which takes the charge of all motion integration and other computations. To perform efficient computation on FPGA, we present two methods, one is combining discrete function and interpolation for computing high power, and the other is using Filter filtrate particles and exploiting two LJ force Calculators.
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