Limit cycles of a class of Liénard systems derived from state-dependent impulses

Discontinuous phenomena, in which objects may behave continuously and sometimes discretely are not only found in nature and under laboratory conditions but also in simple, familiar contexts. For example, this phenomenon is skillfully incorporated into the internal structure of mechanical wristwatches. Unless an extremely small amount of state-dependent impulse is applied intermittently, the reciprocating rotational movement of the balance and hairspring, which is the heart of the mechanical wristwatch, cannot be maintained. The small amount of state-dependent impulse, which is often overlooked, can make a significant difference; however, very few studies have examined this subject. This study assumes the underlying cause of discontinuous behaviors as impulses generated when an object reaches a particular state, assuming that the continuous behavior follows the Liénard system, which is widely studied in the field of electrical circuits. The main theorem provides the conditions under which the effect of the impulses causes a stable limit cycle in the Liénard system, even if no limit cycle exists when there are no impulses. The Poincaré–Bendixson theorem for discontinuous dynamical systems and phase plane analysis are used to prove the main theorem. Several examples and their simulations are provided to illustrate the main theorem.     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

Cyclical games with prohibitions

We consider a certain combinatorial game on a digraph for two cases of the price function. For one case the game in question extends the cyclical game studied in Ehrenfeucht and Mycielski (1979) and Gurvitch, Karzanov and Khachiyan (1988) which, in its turn, is a generalization of the well-known problem of finding a minimum mean cycle in an edge-weighted digraph. We prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies.This work was partially supported by Chaire municipale, Mairie de Grenoble.     

土壤环境中铅的化学

论述了环境中铅的背景值、来源、污染、质量标准、化学循环以及环境中铅的烷基化，为铅污染治理提供理论依据。     

Fe_3(CO)_9(μ_3-S)_2的P(SR)Cl_2和P(SR′)_3取代物的合成和结构研究

用P(SR)Cl_2和P(SR′)_3作n-酸配体,分别与Fe_3(CO)_9(μ_3-S)_2进行取代反应,得到6种新的一取代产物Fe_3(CO)_8(μ_3-S)_2L和3种新的二取代产物Fe_3(CO)_7(μ_3-S)_2L_2,并对它们进行了IR、~1H NMR和MS表征,测定了其中一种取代物Fe_3(CO)_8(μ_3-S)_2[P(SC_6H_5)Cl_2]的分子和晶体结构。     

Thermodynamic investigation of zirconium diselenite

Summary On the basis of calorimetric research of selenium dioxide, zirconium dioxide and zirconium diselenite dissolution reactions in the hydrofluoric acid solution under 298 K a standard enthalpy of Zr(SeO₃)₂ formation reaction from ZrO₂ and SeO₂ and a standard enthalpy of zirconium diselenite formation have been obtained. The value of enthalpy has been equal to -58.1±3.43 kJ mol^-1 in ZrO_2(solid)+2SeO_2(solid) Zr(SeO₃)_2(solid) reaction. The standard enthalpy of zirconium diselenite formation is equal to H_f,298⁰Zr(SeO₃)_2(solid)= -1603.2±3.8 kJ mol^-1. The H_f,298⁰ Zr(SeO₃)_2(solid) value has been determined for the first time.     

Organomineral sorbents based on clinoptilolite-containing tuffs

The anion-exchange properties of a new organomineral sorbent obtained by modification of clinoptilolite-containing tuffs by polyhexamethyleneguanidine have been studied after different periods of storage (time after synthesis) and numbers of sorption-regeneration cycles. The sorbent can be used as a cation- and an anion-exchanger simultaneously. Selectivity coefficients (exchange constants) for F^–, SO₄ ^2–, and HPO₄ ^2– ionsvs. Cl^– ions on modified clinoptilolite-containing tuffs have been determined. The modification improves the mechanical properties of clinoptilolite tuffs.For part 1, see Ref. 1.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1554–1556, September, 1994.     

Supramolecular framework adducts constructed of hexamethylenetetramine,aquated alkali metal cationic clusters,and hexacyanoferrates(II/III)

The reaction of alkali metal hexacyanoferrate(II/III) with (CH₂)₆N₄ (hexamethylenetetramine, abbreviated HMT) in an acidic medium yielded crystalline compounds of stoichiometries HK₂[Fe¹¹¹(CN)₆]·2HMT·4H₂O, H₂K₂[Fe¹¹(CN)₆]·2HMT·4H₂O, and HNa₂[Fe¹¹¹(CN)₆]· 2HMT·5H₂O. Their crystal structures are based on a packing of three molecular components: neutral and/orprotonated HMT, hexacyanoferrate, and an alkali metal ion-water cluster. The resulting three-dimensional supramolecular framework is constructed from the coordination of the alkali metal ion by aqua ligands as well as [Fe(CN)₆]{n–} and HMT units, and further stabilization is achieved by hydrogen bonding between water molecules and the noncoordinated nitrogen atoms of HMT and hexacyanoferrate.     

Synthesis and structural characterization of a coordination polymer of silver(I) with saccharinate and nicotinamide

A silver(I)-saccharinato (sac) complex with nicotinamide (nia), [Ag(sac)(nia)] _n has been synthesized and characterized by elemental analysis, IR spectroscopy, DTA-TG analyses and single crystal X-ray diffractometry. The complex crystallizes in monoclinic space group P2₁/n with unit cell parameters of a=7.0258(4) Å, b=24.3784(10) Å, c=8.4301(5) Å, β=109.407(5)°, V=1361.85(13) ų and Z=4. [Ag(sac)(nia)] _n contains [Ag(sac)(nia)] units, which are doubly bridged by both nia and sac ligands, leading to a linear one-dimensional polymeric chains running along the a axis. The silver(I) ion has a highly distorted AgN₂O₂ tetrahedral geometry and the coordination polymer exhibits relatively short intra-chain ligand supported Ag···Ag separations of 3.1593(4) Å. The one-dimensional chains are crosslinked by N–H···O hydrogen bonds and aromatic π(sac)···π(nia) stacking interactions to generate a two-dimensional layer structure. IR spectra and thermal analysis data are in agreement with the crystal structure.     

Enhanced visible light photocatalytic activity of CeO2@Zn0.5Cd0.5S by facile Ce(IV)/Ce(III) cycle

In this study, CeO₂@Zn_0.5Cd_0.5S heterostructure (Ce@ZCS) is synthesized via a simple two-step hydrothermal method. The effect of CeO₂ loading on the visible-light photoactivity of Zn_0.5Cd_0.5S is mainly investigated. It is found that Ce@ZCS shows a 1.9 times activity as high as ZCS for the MB degradation. The improved activity mainly results from the significant enhanced charge separation by CeO₂, in which the electron transfer is obviously promoted by the facile Ce(IV)/Ce(III) cycle. The excited electrons of ZCS is easy to transfer to CeO₂, thus obviously increasing the charge separation of ZCS. The accepted electrons by CeO₂ may easily be captured by the adsorbed O₂ to form O₂⁻, and then O₂⁻ could combine with H⁺/H₂O to form HO₂, and OH. Finally, O₂⁻, h⁺ and OH are confirmed as the major oxidative species in photocatalytic reaction for Ce@ZCS, and a possible photocatalytic mechanism is proposed. The cheap, efficient Ce@ZCS photocatalyst could be applied for practical waste water treatment.