Inverse Element Research Articles

Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations

In this paper, we introduce the concepts of Bochner and Bohr almost automorphic functions on the semigroup induced by complete-closed time scales and their equivalence is proved. Particularly, when \begin{document}$ \Pi = \mathbb{R}^{+} $\end{document} (or \begin{document}$ \Pi = \mathbb{R}^{-} $\end{document} ), we can obtain the Bochner and Bohr almost automorphic functions on continuous semigroup, which is the new almost automorphic case on time scales compared with the literature [ 20 ] (W.A. Veech, Almost automorphic functions on groups, Am. J. Math., Vol. 87, No. 3 (1965), pp 719-751) since there may not exist inverse element in a semigroup. Moreover, when \begin{document}$ \Pi = h\mathbb{Z}^{+},\,h>0 $\end{document} (or \begin{document}$ \Pi = h\mathbb{Z}^{-},\,h>0 $\end{document} ), the corresponding automorphic functions on discrete semigroup can be obtained. Finally, we establish a theorem to guarantee the existence of Bochner (or Bohr) almost automorphic mild solutions of dynamic equations on semigroups induced by time scales.

Informal Complete Metric Space and Fixed Point Theorems

The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal vector space can also be endowed with a metric to define a so-called informal metric space. The completeness of informal metric space can be defined according to the similar concept of a Cauchy sequence. A new concept of fixed point and the related results are studied in informal complete metric space.

Informal Norm in Hyperspace and Its Topological Structure

The hyperspace consists of all subsets of a vector space. Owing to a lack of additive inverse elements, the hyperspace cannot form a vector space. In this paper, we shall consider a so-called informal norm to the hyperspace in which the axioms regarding the informal norm are almost the same as the axioms of the conventional norm. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets. In this case, the topologies generated by these different concepts of open sets are investigated.

The properties of the ordered bilinear form semigroup related to their regular elements

A semigroup is an algebraic structure which only involve one binary operation that is associative. Research on a semigroup has grown rapidly, included an ordered semigroup. An ordered semigroup is a semigroup that is added an ordered operation such that it meets some axioms. A bilinear forms semigroup is a semigroup whose the elements are adjoin ordered pairs relative to a bilinear form. In this study, we have investigated the properties of an ordered bilinear form semigroup based on its regular elements. We have obtained some properties associated with the biggest inverse element and the biggest idempotent element. The other results we investigated the properties of Right (Left) Green equivalence classes on the ordered bilinear form semigroup related to their biggest inverse element and the biggest idempotent element.

Sound field reconstruction using inverse boundary element method and sparse regularization.

The inverse boundary element method (IBEM) is a powerful tool for realizing sound field reconstruction of sources with arbitrarily-shaped surfaces. In the conventional IBEM, the Tikhonov regularization is generally used and the number of sampling points is required to be larger than that of nodes on the boundary surface to guarantee to obtain a unique solution. Meanwhile, it requires that the minimum discretization interval on the boundary surface should be less than one-sixth wavelength to ensure to obtain enough calculation accuracy. Therefore, the number of sampling points may be dramatically large at high frequencies. In this paper, acoustic radiation modes, which are composed of the eigenvectors of the resistive impedance matrix, are used as the sparse basis of source surface velocities. Based on this sparse basis, sparse regularization is introduced into the IBEM. Compared to the Tikhonov regularization, the sparse regularization can provide a higher accuracy for the reconstruction of source surface velocities and can reduce the number of sampling points by taking advantage of the theory of compressive sensing. Both numerical simulation and experimental results demonstrate the superiority of the proposed method. Meanwhile, the effects of the number of sampling points and the signal-to-noise ratio on the reconstruction accuracy are analyzed numerically.

Comparisons of regularization methods and regularization parameter selection methods in sound source identification using inverse boundary element method

In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further verified through the reconstruction of the vibration of an aluminum plate. Additionally, to reduce the large errors at some frequencies in the reconstruction result, increasing the number of measuring points is more effective than reducing the distance between the measurement plane and the sound source.

Real-time pure shift measurements for uniformly isotope-labeled molecules using X-selective BIRD homonuclear decoupling.

We introduce a novel selective inversion element for chunked homonuclear decoupling that combines isotope selection via BIRD-filtering with band-selective inversion on the X-heteronucleus and allows efficient real-time decoupling of homonuclear and heteronuclear couplings. It is especially suitable for uniformly isotope-labeled compounds. We discuss in detail the inversion element based on band-selective refocusing on the X-nuclei (BASEREX), highlighting in particular the role of appropriate band-selective shaped refocusing pulses and the application of broadband X-pulses for an effective BIRDd element during homodecoupling. The approach is experimentally verified and studied in detail using uniformly 13C-labeled glucose and a uniformly 15N,13C-labeled amino acid mixture.

Reconstruction of transient pressure and acceleration over a tire surface using the inverse time domain boundary element method

The inverse time domain boundary element method (ITBEM) that is derived from the direct time domain boundary element method by eliminating the retarded time is able to reconstruct the transient pressure and flux on the surface of an arbitrarily shaped source by measuring the pressure on a hologram surface. In the present work, the ITBEM is applied to reconstruct the transient pressure and acceleration over the surface of a tire which is supported away from the ground in a semi-anechoic chamber. The tire is impacted by a rigid sphere to generate a transient sound field, and the measurement is controlled by a trigger which is connected to an acceleration sensor stuck on the surface of the tire. The pressure and acceleration on the surface of the tire are reconstructed from the holographic pressure measured by array microphones. By visualizing the pressure and acceleration with respect to the elapsed time, the wave propagation phenomenon of the pressure and acceleration on the surface of the tire is shown clearly. The comparison of the reconstructed surface acceleration to the measured one demonstrates the effectiveness of ITBEM for transient sound field reconstruction.

Hom-groups, representations and homological algebra

Studying basics of Hom-groups, we introduce two types of modules over a Hom-group G. To find out more about these modules, we introduce Hom-group (co)homology with coefficients in these modules. Despite the associative case we observe that the coefficients of Hom-group homology is different from the ones for Homgroup cohomology. We show that the inverse elements provide a relation between Hom-group (co)homology with coefficients in right and left G-modules. It will be shown that our (co)homology theories for Hom-groups with coefficients could be reduced to the Hochschild (co)homologies of Hom-group algebras. For certain coefficients the functoriality of Hom-group (co)homology will be shown.

Magnetization transfer from protons to quadrupolar nuclei in solid-state NMR using PRESTO or dipolar-mediated refocused INEPT methods.

In solid-state NMR spectroscopy, the through-space transfer of magnetization from protons to quadrupolar nuclei is employed to probe proximities between those isotopes. Furthermore, such transfer, in conjunction with Dynamic Nuclear Polarization (DNP), can enhance the NMR sensitivity of quadrupolar nuclei, as it allows the transfer of DNP-enhanced 1H polarization to surrounding nuclei. We compare here the performances of two approaches to achieve such transfer: PRESTO (Phase-shifted Recoupling Effects a Smooth Transfer of Order), which is currently the method of choice to achieve the magnetization transfer from protons to quadrupolar nuclei and which has been shown to supersede Cross-Polarization under Magic-Angle Spinning (MAS) for quadrupolar nuclei and D-RINEPT (Dipolar-mediated Refocused Insensitive Nuclei Enhanced by Polarization Transfer) using symmetry-based SR412 recoupling, which has already been employed to transfer the magnetization in the reverse way from half-integer quadrupolar spin to protons. We also test the PRESTO sequence with R1676 recoupling using 270090180 composite π-pulses as inversion elements. This recoupling scheme, which has previously been proposed to reintroduce 1H Chemical Shift Anisotropy (CSA) at high MAS frequencies with high robustness to rf-field inhomogeneity, has not so far been employed to reintroduce dipolar couplings with protons. These various techniques to transfer magnetization from protons to quadrupolar nuclei are analyzed using (i) an average Hamiltonian theory, (ii) numerical simulations of spin dynamics, and (iii) experimental 1H → 27Al and 1H → 17O transfers in as-synthesized AlPO4-14 and 17O-labelled fumed silica, respectively. The experiments and simulations are done at two magnetic fields (9.4 and 18.8 T) and several spinning speeds (15, 18-24 and 60 kHz). This analysis indicates that owing to its γ-encoded character, PRESTO yields the highest transfer efficiency at low magnetic fields and MAS frequencies, whereas owing to its higher robustness to rf-field inhomogeneity and chemical shifts, D-RINEPT is more sensitive at high fields and MAS frequencies, notably for protons exhibiting large offset or CSA, such as those involved in hydrogen bonds.

A New Concept of Fixed Point in Metric and Normed Interval Spaces

The main aim of this paper is to propose the concept of so-called near fixed point and establish many types of near fixed point theorems in the set of all bounded and closed intervals in R . The concept of null set will be proposed in order to interpret the additive inverse element in the set of all bounded closed intervals. Based on the null set, the concepts of metric interval space and normed interval space are proposed, which are not the conventional metric and normed spaces. The concept of near fixed point is also defined based on the null set. In this case, we shall establish many types of near fixed point theorems in the metric and normed interval spaces.

Performance analysis of a pairwise method for partial inversion of complex block tridiagonal matrices

SummaryThe algorithm detailed below extends previous work on inversion of block tridiagonal matrices from the Hermitian/symmetric case to the general case and allows for varying sub‐block sizes. The sub‐blocks of the matrix are evenly distributed across p processes. Local sub‐blocks are combined to form a matrix on each process. These matrices are inverted locally and the inverses are combined in a pairwise manner. At each combination step, the updates to the global inverse are represented by updating “matrix maps” on each process. The matrix maps are finally applied to the original local inverse to retrieve the block tridiagonal elements of the global inverse. This algorithm has been implemented in Fortran with MPI. Calculated inverses are compared with inverses obtained using the well known libraries ScaLAPACK and MUMPS. Results are given for matrices arising from Density Functional Theory applications.

Near-field acoustic holography with three-dimensional scanning measurements

Data acquisition is one of the key steps in near-field acoustic holography. Most of the existing measurement methods become time-consuming or even impractical when the total number of measurement points is excessively large, for example for the sound field reconstruction of a large-scale source or for a mid-high frequency problem. Alternatively, a three-dimensional scanning near-field acoustic holography (3DSNAH) method is hereby proposed to overcome such problems providing the sound field is time stationary. Data measurement at a large number of points can be accomplished in a matter of minutes by using scanning measurements to acquire the holographic data. The data is then used in an inverse boundary element method to obtain a detailed reconstruction of the sound field radiated by the sound source. Comparison with the traditional point-by-point measurement method demonstrates that the proposed method can acquire holographic data of good quality and reconstruct the sound field with a reasonable accuracy, which verifies the feasibility of the 3DSNAH.

Acoustic source reconstruction and visualization based on acoustic radiation modes

Fourier-based Nearfield Acoustical Holography (NAH), Statistically Optimized Nearfield Acoustical Holography (SONAH) and the Equivalent Source Method (ESM) are widely used in noise source identification and as important tools to guide design modification for noise control purposes. Fourier transform-based NAH requires the sound field to fall to negligible levels outside the measurement aperture, which is a requirement that is rarely met in practice. To overcome this difficulty, SONAH and ESM have been developed. In addition, the Inverse Boundary Element Method (IBEM) can also be used, given sufficient computational resources. Unfortunately, none of these methods can directly guide the design modifications required to unequivocally reduce noise radiation from sources. Previously, radiation mode analysis has been primarily associated with the forward prediction of sound power radiated from noise sources. Since radiation modes contribute independently to the sound power radiation, it is only necessary to modify the surface vibration so that it is not strongly coupled with those modes having high radiation efficiencies in order to ensure sound power reduction. In the current work, an inverse method based on radiation modes was investigated, in which the radiation modes were used as the basis functions to describe surface motion of a source. Thus, this procedure allows the surface vibration that results in the majority of the radiated sound power to be identified unequivocally, and so will, in turn, guide the design changes needed to reduce radiated sound power.

Joint local earthquake and teleseismic inversion for 3-D velocity and Q in New Zealand

We have developed a joint inversion of local earthquake and teleseismic data and applied it in New Zealand to obtain improved 3-D seismic velocity, Qp and Qs models. To accommodate sharper detail where there are dense local earthquake data, varied spatial resolution is enabled with linking of nodes into larger inversion elements where data are sparse. The teleseismic t∗ measurements are obtained by stacking and removing the average for the common source and path. The combined inversions used data from 126 teleseisms and 2365 local earthquakes that extend to 350 km depth. Vp results are similar to previous studies, but have enhanced detail near the slab provided by local seismicity. Sub-slab low velocity and low Q is imaged along the eastern North Island, which may represent asthenosphere with entrained water and melt. Along the western South Island, there is a high Vp, high Q feature distinct from the current slab, and interpreted to be a slab remnant from the period of Subduction-Transform Edge Propagator fault development and trench rollback. The axial South Island high Vp, high Q feature of the Hikurangi Plateau is imaged, and is bounded on the southeast by thinner lithosphere that may be related to the period of Great South Basin rifting.

Analysis of Artifacts Caused by Pulse Imperfections in CPMG Pulse Trains in NMR Relaxation Dispersion Experiments

Nuclear magnetic resonance relaxation dispersion (rd) experiments provide kinetics and thermodynamics information of molecules undergoing conformational exchange. Rd experiments often use a Carr-Purcell-Meiboom-Gill (CPMG) pulse train equally separated by a spin-state selective inversion element (U-element). Even with measurement parameters carefully set, however, parts of 1H–15N correlations sometimes exhibit large artifacts that may hamper the subsequent analyses. We analyzed such artifacts with a combination of NMR measurements and simulation. We found that particularly the lowest CPMG frequency (νcpmg) can also introduce large artifacts into amide 1H–15N and aromatic 1H–13C correlations whose 15N/13C resonances are very close to the carrier frequencies. The simulation showed that the off-resonance effects and miscalibration of the CPMG π pulses generate artifact maxima at resonance offsets of even and odd multiples of νcpmg, respectively. We demonstrate that a method once introduced into the rd experiments for molecules having residual dipolar coupling significantly reduces artifacts. In the method the 15N/13C π pulse phase in the U-element is chosen between x and y. We show that the correctly adjusted sequence is tolerant to miscalibration of the CPMG π pulse power as large as ±10% for most amide 15N and aromatic 13C resonances of proteins.

Observation of proximities between spin-1/2 and quadrupolar nuclei in solids: Improved robustness to chemical shielding using adiabatic symmetry-based recoupling

We introduce a novel heteronuclear dipolar recoupling based on the R21−1 symmetry, which uses the tanh/tan (tt) shaped pulse as a basic inversion element and is denoted R21−1(tt). Using first-order average Hamiltonian theory, we show that this sequence is non-γ-encoded and that it reintroduces the |m| = 1 spatial component of the Chemical Shift Anisotropy (CSA) of the irradiated isotope and its heteronuclear dipolar interactions. Using numerical simulations and one-dimensional (1D) 27Al-{31P} through-space D-HMQC (Dipolar Heteronuclear Multiple-Quantum Correlation) experiments on VPI-5, we compare the performances of this recoupling to those of other non-γ-encoded |m| = 1 heteronuclear recoupling schemes: REDOR (Rotational-Echo DOuble Resonance), SFAM (Simultaneous Frequency and Amplitude Modulation) and R42−1(tt). Such comparison indicates that the R21−1(tt) scheme is more robust to CSA, offset and radiofrequency field inhomogeneities than the other schemes. We take advantage of the high robustness of R21−1(tt) to CSA and offset to demonstrate the possibility to correlate the signals of 207Pb isotope with those of neighboring half-integer spin quadrupolar nuclei. Such approach is demonstrated experimentally by acquiring 11B-{207Pb} D-HMQC 2D spectra of Pb4O(BO3)2 crystalline powder.

Near Fixed Point Theorems in the Space of Fuzzy Numbers

The fuzzy numbers are fuzzy sets owning some elegant mathematical structures. The space consisting of all fuzzy numbers cannot form a vector space because it lacks the concept of the additive inverse element. In other words, the space of fuzzy numbers cannot be a normed space even though the normed structure can be defined on this space. This also says that the fixed point theorems established in the normed space cannot apply directly to the space of fuzzy numbers. The purpose of this paper is to propose the concept of near fixed point in the space of fuzzy numbers and to study its existence. In order to consider the contraction of fuzzy-number-valued function, the concepts of near metric space and near normed space of fuzzy numbers are proposed based on the almost identical concept. The concepts of Cauchy sequences in near metric space and near normed space of fuzzy numbers are also proposed. Under these settings, the existence of near fixed points of fuzzy-number-valued contraction function in complete near metric space and near Banach space of fuzzy numbers are established.

Near Fixed Point Theorems in Hyperspaces

The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.

Inverse Green element simulations of instantaneous pollutant injections into a 2-D aquifer

When pollution spills occur, they impact on the quality of water in underlying aquifers. Such spills can be modelled as instantaneous pollution sources, and estimating their strengths from the conc...