Periodic solutions of second order nonlinear difference equations with discrete -Laplacian

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete -Laplacian. We obtain in particular upper and lower solutions theorems, Ambrosetti–Prodi type multiplicity results, sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0.     

On the spectral analysis of quantum field Hamiltonians

We define C^∗-algebras on a Fock space such that the Hamiltonians of quantum field models with positive mass are affiliated to them. We describe the quotient of such algebras with respect to the ideal of compact operators and deduce consequences in the spectral theory of these Hamiltonians: we compute their essential spectrum and give a systematic procedure for proving the Mourre estimate.     

Finite element analysis of an A– method for time-dependent Maxwell’s equations based on explicit-magnetic-field scheme

In this paper, error analysis of a finite element A– method for the time-dependent Maxwell’s equations is presented. An explicit-magnetic-field scheme is applied. Provided that the time-stepsize τ is sufficiently small, the proposed algorithm yields for finite time T an error of in the L²-norm for the electric field E and the magnetic field H, where h is the mesh size.     

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

It is proved that the Stokes operator in L^q -space on an infinite cylindrical domain of , , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L ^q where and is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on -boundedness of operator families and on square function estimates involving Muckenhoupt weights.     

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

Synthesis,Crystal Structure and Spectroscopic Properties of 1,2-Benzothiazine Derivatives:An Experimental and DFT Study

1,2-Benzothiazine derivatives methyl 3-methoxy-4-oxo-3,4-dihydro-2H-benzo[e] [1,2]thiazine-3-carboxylate 1,1-dioxide(1) and methyl 2-ethyl-3-hydroxy-4-oxo-3,4-dihydro-2Hbenzo[e][1,2]thiazine-3-carboxylate 1,1-dioxide(2) were synthesized, and characterized by spectroscopic techniques; 1H-NMR and infrared(IR) spectroscopy. Crystals of 1 and 2 were grown by slow evaporation of methanol and ethyl acetate, respectively and their crystal structures were investigated by single-crystal X-ray diffraction analysis. Geometric properties were calculated by the B3 LYP method of density functional theory(DFT) at the 6-31G+(d) basis set to compare with the experimental data. Simulated properties were found in strong agreement with the experimental ones. Intermolecular forces have also been modeled in order to investigate the strength of packing and strong hydrogen bonding was observed in both compounds 1 and 2. Electronic properties such as Ionization Potential(IP), Electron Affinities(EA) and coefficients of the highest occupied molecular orbital(HOMO) and the lowest unoccupied molecular orbital(LUMO) of com- pounds 1 and 2 were simulated for the first time.     

An Efficient Synthesis of Chromeno[2,3‐d]pyrimidine‐2,4‐diones with a Nitroketene‐Aminal Moiety at C(5) by a One‐Pot Four‐Component Reaction

An efficient and novel synthesis of chromeno[2,3‐d]pyrimidine‐2,4‐dione derivatives with a nitroketene‐aminal moiety at C(5) via four‐component reaction of salicylaldehydes, barbituric acid, diamines, and 1,1‐bis(methylsulfanyl)‐2‐nitroethene in EtOH and in the presence of AcOH is reported. Easy performance, good yields, and easy purification are the main advantages of this method. All structures were confirmed by IR, MS, and ¹H‐ and ¹³C‐NMR, and by X‐ray crystal‐structure analyses. A plausible mechanism for this type of reaction is proposed (Scheme).     

Synthesis and Spectral Characteristics of Novel Fluorescent Dyes Based on Pyrimido[4,5‐d] [1,2,4]triazolo[4,3‐a]pyrimidine

Different derivatives of a novel heterocyclic system, i.e., pyrimido[4,5‐d] [1,2,4]triazolo[4,3‐a]pyrimidine, are synthesized in moderate‐to‐good yields. These compounds exhibit excellent photochromism upon photoirradiation. The photophysical characterizations of these new compounds were evaluated by UV/VIS absorption and fluorescence emission studies. The emission spectra in various solvents are also presented and discussed. The changes are due to the intramolecular H‐bonding of pyrimido‐triazolo‐pyrimidine with H₂O, and photoinduced electron and general solvent effect. These compounds display high fluorescence quantum yields and are reported as new fluorophores.     

Synthesis and spectroscopic characterization of novel 2-amino-4,5,6,7-tetrahydro-3H-cyclopenta[d]pyrimidine and pyrimido[1,2-a]pyrimidine derivatives

2-Aminocyclopenta[d]pyrimidines 3a-c were achieved via a one-pot, three-component reactions of cyclopentanone 1, aromatic aldehyde and guanidine hydrochloride (1:2:1 molar ratio). Also, cyclization of 2,5-bis-(arylmethylidene)cyclopentanones 2 with guanidine hydrochloride (1:1 molar ratio) in methanol in the presence of sodiummethoxide afforded cyclopenta-[d]pyrimidines 3. Compound 3a has been shown to be a useful building block for the synthesis of some novel pyrimido[1,2-a]pyrimidines 5, 7 and 12. The structures of the newly synthesized compounds were confirmed on the basis of analytical and spectral data.     

Highly Efficient and Air-Stable Lanthanide EuII Complex: New Emitter in Organic Light Emitting Diodes

Luminescent Eu^II complexes with a characteristic 5d–4f transition have potential applications in many fields. However, their instability in ambient conditions impedes further exploration and application. Herein, we report two new Eu^II complexes, bis[hydrotris(3-trifluoromethylpyrazolyl)borate]europium(II) ( Eu-1 ) and bis[hydrotris(3-methylpyrazolyl)borate]europium(II) ( Eu-2 ). Intriguingly, the blue emissive Eu-1 showed high air stability arising from fluorine protection and close molecular packing, as maintaining a photoluminescence quantum yield (PLQY) of 91 % (initial 96 %) upon exposure to air over 2200 hours. While the orange emissive Eu-2 showed a maximum luminance of 30620 cd m⁻², and a maximum external quantum efficiency (EQE) of 6.5 %, corresponding to an exciton utilization efficiency around 100 % in organic light-emitting diodes (OLEDs). These results could inspire further research on stable and efficient Eu^II complexes and their application in OLEDs.