The chaotic dynamic analysis along with chaos controller of an active suspension in vehicles has been studied in this paper. The unstable periodic orbits of the system are stabilized using the developed delay feedback control algorithm based on the fuzzy sliding mode system. Firstly, the equations of motions in the chaotic half-vehicle model are derived via Newton-Euler rules and simulated by the fourth order Runge-Kutta method. Then, Forcing frequency has been used to confirm nonlinear phenomenon such as jump and chaos in the vehicle system. Critical values of the control parameters in the Forcing frequency demonstrate the changes of system behavior from the periodic to the irregular chaotic responses. In order to eliminate the chaotic behaviors in the vertical dynamics of vehicle, a novel fuzzy sliding delay feedback control algorithm is developed on the active suspension with chaotic responses. Using fuzzy logic, the controller gain of the sliding delay feedback control is online estimated that is caused to reject the chattering phenomenon in the sliding mode algorithm beside the improvement of the responses. Simulation results of the control system depict a reduction of settling time and energy consumption along with eliminating the overshoots and chaotic vibrations

The current study aims to establish a connection between graphs and automata theory, which apparently demonstrate di erent mathematical structures. Through searching out some properties of one of these structures, we try to  nd some new properties of the other structure as well. This will result in obtaining some unknown properties. At  rst, a novel automaton called zero-Forcing (Z-F)  nite automata is de ned according to the notion of a zero-Forcing set of a graph. It is shown that for a given graph and for some zero Forcing sets, various Z-F- nite automata will be obtained. In addition, the language and the closure properties of Z-F- nite automata, in particular; union, connection, and serial connection are studied. Moreover, considering some properties of graphs such as the closed trail, connected and complete; some new features for Z-F- nite automata are presented. Further, it is shown that there is not any  nite graph such that f be a part of the language of its Z-F- nite automata. Actually, it is proved that for every given graph, the Z-F- nite automata of it does not show any closed trail containing all edges for every zero Forcing set, but if the graph G has been a closed trail containing all edges, then the Z-F- nite automata of it has a weak closed trail containing all edges. Some examples are also given to clarify these new notions.

Introduction: Usually osseointegration takes between three to six months after implant placement but patients are interested to have early loading. There are no definitive criteria for measuring bone mineral density (BMD), insertion torque (IT) (final torque force) and resonance frequency analysis (RFA) (primary implant stability) to determine exact loading time based on the relationship between the above-mentioned parameters. The aim of this study was to determine the relationship between IT, RFA and BMD in screw-type implants.Materials and Methods: This clinical trial was conducted on 18 patients who were candidates for ITI implant placement. Written consent was taken and jaw bone density was determined via a digital radiography technique before surgery. After implant placement, RFA and IT were measured. Fifty-five ITI implants of the total 62 implants placed were evaluated; the implants were 12 mm long with a diameter of 4.1 mm. Data was analyzed with Pearson’s test using SPSS.15 software (a=0.05).Results: There was a significant relationship between IT, RFA and BMD. Pearson’s test showed a correlation coefficient of 0.872 to 0.789 between the three parameters, indicating a strong relationship between them. The mean bone density was 1.468±0.042 g/cm2; the mean RFA was 66.01±2.2 ISQ and the mean IT was 34.62±3.33 N/cm2.Conclusion: Based on the results of the present study there is a significant relationship between, IT, RFA and BMD (p value=0.001).

The idea of “Forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics [D. Donovan, E. S. Mahmoodian, C. Ramsay, A. P. Street, Defining sets in combinatorics: A survey, in: C. D. Wensley (Ed.), Surveys in Combinatorics, Cambridge Univ. Press, 2003, pp. 115-174]. Recently, the Forcing on perfect matchings has been attracting more researchers’ attention. A Forcing set of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G. A global Forcing set of G, introduced by Vukičvićet al., is a subset of E (G) on which there are distinct restrictions of any two different perfect matchings of G. Combining the above “Forcing” and “global” ideas. Xu et al. in [Complete Forcing numbers of catacondensed benzenoid, J. Combin. Optim.29 (2015) 803-814.] introduced a complete Forcing set of G defined as a subset of E (G) on which the restriction of any perfect matching M of G is a Forcing set of M. The minimum cardinality of complete Forcing sets is the complete Forcing number of G. In this paper, we give the explicit expressions for the complete Forcing number of several classes of polyphenyl systems.

A subset of the vertex set of a graph $G$ is called a zero Forcing set if by considering them colored and, as far as possible, a colored vertex with exactly one non-colored neighbor forces its non-colored neighbor to get colored, then the whole vertices of $G$ become colored. The total Forcing number of a graph $G$, denoted by $F_t(G)$, is the cardinality of a smallest zero Forcing set of $G$ which induces a subgraph with no isolated vertex. The connected Forcing number, denoted by $F_c(G)$, is the cardinality of a smallest zero Forcing set of $G$ which induces a connected subgraph. In this paper, we first characterize the graphs with $F_t(G)=2$ and, as a corollary, we characterize the graphs with $F_c(G)=2$.

Let be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of is a set of disjoint edges which covers all vertices of . The anti-Forcing number of is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by . For every , the th power of , denoted by , is a graph with the same vertex set as such that two vertices are adjacent in if and only if their distance is at most in . In this paper, we study the anti-Forcing number of the powers of some graphs.

‎For a connected graph $G=(V,E)$ of order at least two‎, ‎an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$‎. ‎The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$‎. ‎A subset $T$ of $S$ is a Forcing edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$‎. ‎A Forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum Forcing edge detour monophonic subset of $S$‎. ‎The Forcing edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum Forcing edge detour monophonic subset of $S$‎. ‎The Forcing edge detour monophonic number of $G$ is $f_{edm}(G)=min\{f_{edm}(S)\}$‎, ‎where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$‎. ‎We determine bounds for it and find the Forcing edge detour monophonic number of certain classes of graphs‎. ‎It is shown that for every pair a‎, ‎b of positive integers with $0\leq a

In this paper, for a given graph, G, some domination parameters and the Forcing domination number of the graph, M (G), obtained from G arising in Mycielski's construction, are studied.