A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method allows a surface to be given implicitly as a zero level of a level set function. A surface equation is extended to a narrow-band neighborhood of the surface. The resulting extended equation is a non-degenerate PDE and it is solved on a bulk mesh that is unaligned to the surface. An unfitted finite element method is used to discretize extended equations. Error estimates are proved for finite element solutions in the bulk domain and restricted to the surface. The analysis admits finite elements of a higher order and gives sufficient conditions for archiving the optimal convergence order in the energy norm. Several numerical examples illustrate the properties of the method.Comment: arXiv admin note: text overlap with arXiv:1301.470

Powers of sets in free groups

We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a free group if A contains at least two noncommuting elements, where c_n>0 are constants not depending on A. Simple examples show that the order of these estimates are the best possible for each n>0.Comment: 3 page

Control of the nonlinear frequency shift for the spin-transfer nanooscillator using a bias magnetic field

We investigated the possibilities of controlling the nonlinear frequency shift of the magnetization oscillations in a spin-transfer nanoscillator by varying the magnitude and direction of the bias magnetic field. We considered both isotropic ferromagnetic materials and crystals with uniaxial and cubic crystallographic anisotropies. We have shown that achieving a zero nonlinear frequency shift is possible with a certain orientation of the bias magnetic field vector. The results of the theoretical analysis based on the method of Hamiltonian formalism are in good agreement with the micromagnetic simulations. Our research reveals the way to control the frequency tuning of a spin transfer nanoscillator, which is crucial for spintronic signal generation devices.Comment: 6 pages, 3 figure

Noise Properties of Two Mutually Coupled Spin-Transfer Nanooscillators in the Phase Locking Regime

Introduction. Today, many research endeavors are devoted to the miniaturization of microwave sources. One of the promising approaches is the use of magnetic nanostructures (spintronics elements), providing a wide range of frequency tuning and low power consumption. The main disadvantage of spintronics generators (spintransfer nanoscillators ‒ STNO) is a low output power of generated oscillations (tens of nanowatts and less). A possible solution is to sum up the power of many STNOs in a mutual synchronization mode.Aim. The investigation of noise properties of two connected STNOs with identical and non-identical parameters in a phase synchronization mode.Materials and methods. A model was developed of two STNOs interconnected by spin waves taking into account thermal noises. Spectral power densities of the amplitude and phase noise were obtained by the method of effective linearization.Results. Dependencies were obtained in a general form for attenuation coefficients of the amplitude and phase fluctuations of noise sources for each STNO. Three cases of synchronization were considered: completely identical STNOs, two identical STNOs but with different oscillation frequencies, and two non-identical STNOs, differing in an allowance of self-excitation by frequencies and amplitudes of the oscillations. It was possible to obtain a gain in the amplitude and phase noise for two identical STNOs. In this case, an increase in the allowance of self-excitation led to a decrease in the level of phase and amplitude noise.Conclusion. This analysis of the attenuation coefficients for non-identical STNOs demonstrates the possibility of improving the noise properties of each of the generators. In this case, the best noise value is obtained for an STNO with greater stability in a stand-alone mode.Introduction. Today, many research endeavors are devoted to the miniaturization of microwave sources. One of the promising approaches is the use of magnetic nanostructures (spintronics elements), providing a wide range of frequency tuning and low power consumption. The main disadvantage of spintronics generators (spintransfer nanoscillators ‒ STNO) is a low output power of generated oscillations (tens of nanowatts and less). A possible solution is to sum up the power of many STNOs in a mutual synchronization mode.Aim. The investigation of noise properties of two connected STNOs with identical and non-identical parameters in a phase synchronization mode.Materials and methods. A model was developed of two STNOs interconnected by spin waves taking into account thermal noises. Spectral power densities of the amplitude and phase noise were obtained by the method of effective linearization.Results. Dependencies were obtained in a general form for attenuation coefficients of the amplitude and phase fluctuations of noise sources for each STNO. Three cases of synchronization were considered: completely identical STNOs, two identical STNOs but with different oscillation frequencies, and two non-identical STNOs, differing in an allowance of self-excitation by frequencies and amplitudes of the oscillations. It was possible to obtain a gain in the amplitude and phase noise for two identical STNOs. In this case, an increase in the allowance of self-excitation led to a decrease in the level of phase and amplitude noise.Conclusion. This analysis of the attenuation coefficients for non-identical STNOs demonstrates the possibility of improving the noise properties of each of the generators. In this case, the best noise value is obtained for an STNO with greater stability in a stand-alone mode

Copper(I) Complexes of N-thiophosphorylated Bis-thiourea [CH2NHC(S)NHP(S)(OiPr)2]2 and Phosphines (PPh3, Ph2P(CH2)1–3PPh2, Ph2P(C5H4FeC5H4)PPh2): Versatile Structures and Luminescence

Reaction of the potassium solution of [CH2NHC(S)NHP(S)(OiPr)2]2 (H2L) with [Cu(PPh3)3I] or a mixture of CuI and Ph2P(CH2)1–3PPh2 or Ph2P(C5H4FeC5H4)PPh2 in aqueous EtOH/CH2Cl2 leads to binuclear [Cu2(PPh3)2L] (1), [Cu2(Ph2PCH2PPh2)L] (2), [Cu2{Ph2P(CH2)2PPh2}2L] (3), [Cu2{Ph2P(CH2)3PPh2}2L] (4) or [Cu2{Ph2P(C5H4FeC5H4)PPh2}2L] (5) complexes. The structures of these compounds were investigated by IR, 1H and 31P{1H} NMR spectroscopy; their compositions were examined by microanalysis. The luminescent properties of complexes 1–5 in the solid state are reported

Copper(I) Complexes of N-thiophosphorylated Bis-thiourea [CH2NHC(S)NHP(S)(OiPr)2]2 and Phosphines (PPh3, Ph2P(CH2)1–3PPh2, Ph2P(C5H4FeC5H4)PPh2): Versatile Structures and Luminescence

Reaction of the potassium solution of [CH2NHC(S)NHP(S)(OiPr)2]2 (H2L) with [Cu(PPh3)3I] or a mixture of CuI and Ph2P(CH2)1–3PPh2 or Ph2P(C5H4FeC5H4)PPh2 in aqueous EtOH/CH2Cl2 leads to binuclear [Cu2(PPh3)2L] (1), [Cu2(Ph2PCH2PPh2)L] (2), [Cu2{Ph2P(CH2)2PPh2}2L] (3), [Cu2{Ph2P(CH2)3PPh2}2L] (4) or [Cu2{Ph2P(C5H4FeC5H4)PPh2}2L] (5) complexes. The structures of these compounds were investigated by IR, 1H and 31P{1H} NMR spectroscopy; their compositions were examined by microanalysis. The luminescent properties of complexes 1–5 in the solid state are reported

Metal ion influences distortion of the ligand in the structure of [M{2−MeO(O)CC6H4NHC(S)NP(S)(OiPr)2}2](M=ZnII,CdII)[M\{2-MeO(O)CC_6H_4NHC(S)NP(S)(OiPr)_2\}_2] (M = Zn^{II}, Cd^{II}) complexes : a driving force for intermolecular aggregation

Reaction of the in situ deprotonated N-thiophosphorylated thiourea $2-MeO(O)CC_6H_4NHC(S)NHP(S)(OiPr)_2 (HL)$ with $MCl_2 (M = Zn^{II}, Cd^{II})$ in aqueous ethanol leads to complexes of the formula $[ML_2]$. Both compounds crystallise in the triclinic space group P[1 with combining macron] with Z = 2 and the metal cations are found in a tetrahedral S_2S′_2 coordination environment formed by the C–S and P–S sulfur atoms. The crystal structures reveal intramolecular N–H⋯O[double bond, length as m-dash]C hydrogen bonds formed within the $2-MeO(O)CC_6H_4NH$ fragments. Both structures are further stabilised by intermolecular π⋯π stacking interactions, which are more efficient in $[CdL_2]$. Here, a pronounced dimeric intermolecular aggregate is observed which goes along with a pronounced distortion of the chelate [(S)CNP(S)]− backbone of the ligand upon coordination to $Cd^{II}$ as well as a significantly distorted coordination tetrahedron $CdS_2S′_2$. The aggregation is also reflected in the positive electrospray ionisation (ESI) mass spectrum of the CdII complex, which exhibits peaks for the dimeric cations $[Cd_2L_3]+$, $[Cd_2L_4 + H]+$ and $[Cd_2L_4 + Na]+$, while for the ZnII analogue only monomeric species were observed. Quantum chemical ETS-NOCV (ADF) calculations confirm the higher stability of dimers in $[CdL_2]$ compared with $[ZnL_2]$. The π⋯π stacking interactions are prodominantly due to dispersion contributions, though the electrostatic and orbital interaction components are also important. QTAIM (ADF) type calculations additionally quantify the covalent and non-covalent interactions in the momomers