Topological entropy of a graph map

Let G be a graph and f : G → G be a continuous map. Denote by h(f), P(f), AP(f), R(f)and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper,we show that the following statements are equivalent:(1) h(f) 0.(2) There exists an x ∈ G such that ω(x, f) ∩ P(f) = ? and ω(x, f) is an infinite set.(3) There exists an x ∈ G such that ω(x, f)contains two minimal sets.(4) There exist x, y ∈ G such that ω(x, f)-ω(y, f) is an uncountable set and ω(y, f) ∩ω(x, f) = ?.(5) There exist an x ∈ G and a closed subset A ? ω(x, f) with f(A) ? A such that ω(x, f)-A is an uncountable set.(6) R(f)-AP(f) = ?.(7) f |P(f)is not pointwise equicontinuous.     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

Synthesis,Characterization and Antibacterial Activity of <Emphasis Type="Italic">N</Emphasis>-(2,3,4-Trimethoxy-6-methylbenzylidene)-2-methyl-benzenamine

Abstract  The title compound N-(2,3,4-trimethoxy-6-methylbenzylidene)-2-methyl-benzenamine (C₁₈H₂₁NO₃, M _r = 299.36) was synthesized and characterized by elemental analysis, IR spectra and single crystal X-ray diffraction. The crystal belongs to monoclinic, space group P2₁/n, with a = 7.2804(5), b = 8.5909(9), c = 26.117(3) ?, β = 92.056(2)°, V = 1,632.4(3) ?³, Z = 4, D _c = 1.218 g cm⁻³, λ = 0.71073 ?, μ(Mo Kα) = 0.083 mm⁻¹, F(000) = 640. The final refinement gave R = 0.0497, wR(F ²) = 0.1157 for 2,879 observed reflections with I > 2σ(I). X-ray diffraction analysis reveals that the dihedral angle between the two phenyl rings is 25.9 (2)°. The molecule adopts a trans configuration about the central C=N functional bond. The crystal structure is stabilized by C–H…O hydrogen bonds and π–π stacking interactions. The title compound possesses moderate antibacterial activity. Graphical Abstract  The title compound N-(2,3,4-trimethoxy-6-methylbenzylidene)-2-methyl-benzenamine (C₁₈H₂₁NO₃, M _r = 299.36) was synthesized and characterized by elemental analysis, IR spectra and single crystal X-ray diffraction. The crystal belongs to monoclinic, space group P2₁/n, with a = 7.2804(5), b = 8.5909(9), c = 26.117(3) ?, β = 92.056(2)°, V = 1,632.4(3) ?³, Z = 4, D _c = 1.218 g cm⁻³, λ = 0.71073 ?, μ(Mo Kα) = 0.083 mm⁻¹, F(000) = 640. The final refinement gave R = 0.0497, wR(F ²) = 0.1157 for 2,879 observed reflections with I > 2σ(I). X-ray diffraction analysis reveals that the dihedral angle between the two phenyl rings is 25.9 (2)°. The molecule adopts a trans configuration about the central C=N functional bond. The crystal structure is stabilized by C–H…O hydrogen bonds and π–π stacking interactions. The title compound possesses moderate antibacterial activity.      

Preparation and Electrochemical Properties of 3-Phenyl-6-aryl-1,2,4-triazolo[3,4-b][1,3,4]-thiadiazoles

Introduction Since Tang[1] firstly reported the electroluminescence of 8-hydroxyquinoline aluminium, much attention has been paid to the organic compound as electro-luminescent material. Especially, electron transport materials play an important role in organic light-emitting diode(OLED). 2-(4-Biphenyl)-5-(t-butylphenyl)-1,3,4-oxadiazole(PBD), as a kind of popular electron injection/transport materials, can improve remarkably the injection balance of electrons and holes. As a result, it can amend the properties of OLED[2,3].     

关于素环平方封闭的李理想的两个结果

Let R be a 2-torsion free prime ring, d1 a nonzero derivation, -γ a generalized derivation associated with a nonzero derivation d2, U a square closed Lie ideal of R. In the present paper,we prove that if [di^2（u）, u] ∈ Z（R） or γ acts as a homomorphism （or an antihomomorphism） on U, then U Z（R）.     

双频超声辅助酶解提取-石墨炉原子吸收光谱法测定玉米中镉和铅

建立了双频超声辅助酶解提取(DUED)对玉米中Cd(Ⅱ)和Pb(Ⅱ)进行提取的方法,并结合石墨炉原子吸收光谱法(GFAAS)对其检测.结果表明,双频超声辅助酶解提取方法的最佳酶解条件为:0.1 mLα-淀粉酶+0.1 g中性蛋白酶+0.2 g木瓜蛋白酶、2.5 mL提取剂(pH 5.0)和60℃的提取温度.最佳超声参数...     

A Nickel(II) Nitrite Based Molecular Perovskite Ferroelectric

The X‐site ion in organic–inorganic hybrid ABX₃ perovskites (OHPs) varies from halide ion to bridging linkers like HCOO^?, N₃^?, NO₂^?, and CN^?. However, no nitrite‐based OHP ferroelectrics have been reported so far. Now, based on non‐ferroelectric [(CH₃)₄N][Ni(NO₂)₃], through the combined methodologies of quasi‐spherical shape, hydrogen bonding functionality, and H/F substitution, we have successfully synthesized an OHP ferroelectric, [FMeTP][Ni(NO₂)₃] (FMeTP=N‐fluoromethyl tropine). As an unprecedented nitrite‐based OHP ferroelectric, the well‐designed [FMeTP][Ni(NO₂)₃] undergoes the ferroelectric phase transition at 400 K with an Aizu notation of 6/mmmFm, showing multiaxial ferroelectric characteristics. This work is a great step towards not only enriching the molecular ferroelectric families but also accelerating the potential practical applications.     

Phenothiazine Scope: Steric Strain Induced Planarization and Excimer Formation

Phenothiazine derivatives based on the 10‐phenyl‐10H‐phenothiazine ( NAS ) chromophore, namely 7‐phenyl‐7H‐benzo[c]phenothiazine ( NAS‐1 ) and 12‐phenyl‐12H‐benzo[a]phenothiazine ( NAS‐2 ), were designed and synthesized. NAS‐1 and NAS‐2 are constitutional isomers with different steric strains imposed on the phenothiazine core moiety. In solution, the more‐strained NAS‐2 possesses a bent structure and undergoes photoinduced structural planarization (PISP). In the crystal, despite the absence of PISP, bent NAS‐2 exhibits prominent excimer emission as well as emission mechanochromism, which is not observed in the planar‐like NAS and NAS‐1 . This unconventional observation results from the bent core structure facilitating π–π stacking of the peripheral naphthalene moieties. Two‐photon‐coupled depth‐dependent emission shows spectral differences between the surface and kernel of the NAS‐2 crystal, and is believed to be a general phenomenon, at least in part, for materials exhibiting emission mechanochromism.     

Some remarks on one-sided regularity

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.     

New Sesquiterpenoids from Salvia castanea Diels f. tomentosa

Two new sesquiterpenoids, castanin A (= 4R*,5R*,5aS*,8S*,8aR*,8bR*)‐8‐formyl‐4,5,5a,6,7,8,8a,8b‐octahydro‐5,8b‐dihydroxy‐3,5a,8‐trimethyl‐2‐oxo‐2H‐indeno[4,5‐b]furan‐4‐yl acetate; 1 ) and castanin B (= 1aS*,6R*,6aS*,7aR*,9aR*)‐1a,2,6,6a,7a,8,9,9a‐octahydro‐1a,5,7a‐trimethylbisoxireno[4,5 : 8,9]cyclodeca[1,2‐b]furan‐6‐yl acetate; 2 ) were isolated from the aerial parts of Salvia castanea Diels f. tomentosa, together with two known compounds, oplophanone and 1β,6α‐dihydroxyeudesm‐4(14)‐ene. Their structures were elucidated by spectroscopic analyses and, in the case of 2 , also by X‐ray crystallography. Castanin A ( 1 ) represents a novel sesquiterpenoid, with a contracted ring A, derived from eudesmanolide. A possible biogenetic pathway is proposed.