Symmetric Basis Research Articles

Two-dimensional light-frontϕ4theory in a symmetric polynomial basis

We study the lowest-mass eigenstates of $\phi^4_{1+1}$ theory with both odd and even numbers of constituents. The calculation is carried out as a diagonalization of the light-front Hamiltonian in a Fock-space representation. In each Fock sector a fully symmetric polynomial basis is used to represent the Fock wave function. Convergence is investigated with respect to the number of basis polynomials in each sector and with respect to the number of sectors. The dependence of the spectrum on the coupling strength is used to estimate the critical coupling for the positive-mass-squared case. An apparent discrepancy with equal-time calculations of the critical coupling is resolved by an appropriate mass renormalization.

Novel symmetric diimine-Schiff bases and asymmetric triimine-Schiff bases as chemosensors for the detection of various metal ions

In this study, two symmetric diimine-Schiff bases (D1, D2) containing nitro group were synthesized by a simple one-pot condensation of 4-nitro-o-phenylenediamine with substituted-salicylaldehyde (5-Cl, 5-CH3) in 1:2 ratio. After the selective reduction of nitro group to amino group by using sodium dithionite and forming the new imine bond by adding substituted-salicylaldehyde or 2-hydroxy-1-naphthaldehyde, four asymmetric triimine-Schiff bases (T1s, T1n, T2s, and T2n) were obtained. Results of the newly synthesized compounds established by elemental analyses, FT-IR, UV–vis, 2D NMR (HMQC), 1H-NMR, 13C-NMR and TOF-mass spectroscopic experiments were consistent with their chemical structures. The tautomeric equilibria were also studied. The sensor properties of all Schiff bases were examined upon addition of the metal ions, such as Cr3+, Fe2+, Fe3+, Co2+, Ni2+, Cu2+, Zn2+, and Pb2+. The interactions between receptors and ions are easily monitored by UV–vis method. The receptor D2 showed colour changes from yellow to intense deep orange colour for Cu2+, a orange colour for Co2+ and dark yellow colour for other ions. Although metal ions caused no change in colour of T2s, the main absorption band of receptor shifted from 351 nm to 343–372 nm T2n underwent colour changes from yellow to light yellow on gradual addition of Fe3+.

Construction of multiscaling functions using the inverse representation theorem of matrix polynomials

Wavelet analysis deals with finding a suitable basis for the class of $$L^2$$ functions. Symmetric basis functions are very useful in various applications. In the case of all wavelets other than the famous Haar wavelet, the simultaneous inclusion of compact supportedness, orthogonality and symmetricity is not possible. Theory of multiwavelets assumes significance since it offers orthogonal, compact frames without losing symmetry. We can also construct symmetric, compactly supported and pseudo-biorthogonal bases which are also possible only in the case of multiwavelets. The properties of a multiwavelet directly depends on the corresponding multiscaling function. A multiscaling function is characterized by a unique symbol function, which is a matrix polynomial in complex exponential. A matrix polynomial can be constructed from its spectral data. Our aim is to find the necessary as well as sufficient conditions a spectral data must satisfy so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function $$\varPhi (x)$$ . We will construct such a multiscaling function $$\varPhi (x)$$ and its dual $$\tilde{\varPhi }(x)$$ such that the functions $$\varPhi (x)$$ and $$\tilde{\varPhi }(x)$$ form a pair of pseudo-biorthogonal multiscaling functions.

Stable Simplex Spline Bases for $$C^3$$ C 3 Quintics on the Powell–Sabin 12-Split

For the space of $$C^3$$ quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the $$L_\infty $$ norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an $$h^2$$ bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive $$C^0$$ , $$C^1$$ , $$C^2$$ , and $$C^3$$ conditions on the control points of two splines on adjacent macrotriangles.

Effect of Substitutions on Thermal Behavior of Symmetric Doubleschiff Bases of 1,1’-Bis(4-Amino Phenyl)Cyclohexane

Thermo gravimetric analyses of some symmetric double Schiff bases of 1, 1’- bis (4-amino phenyl) cyclohexane (SDSB-1 to SDSB-6) was carried out at 100C/min in nitrogen atmosphere. Schiff bases are thermally stable up to 200-285°C and followed two step degradation kinetic. Various kinetic parameters such as order of degradation (n), energy of activation (Ea), frequency factor (A) and entropy change (ΔS) were derived accordingly Anderson-Freeman method interpreted in light of nature of the substituents. The nature, size and position of substituents affected thermal properties of the Schiff bases.

A New Class of Hilbert Pairs of Almost Symmetric Orthogonal Wavelet Bases

A New Class of Hilbert Pairs of Almost Symmetric Orthogonal Wavelet Bases

Effect of Substitutions on Thermal Behavior of Symmetric Doubleschiff Bases of 1,1’-Bis(4-Amino Phenyl)Cyclohexane

Thermo gravimetric analyses of some symmetric double Schiff bases of 1, 1’- bis (4-amino phenyl) cyclohexane (SDSB-1 to SDSB-6) was carried out at 100C/min in nitrogen atmosphere. Schiff bases are thermally stable up to 200-285°C and followed two step degradation kinetic. Various kinetic parameters such as order of degradation (n), energy of activation (Ea), frequency factor (A) and entropy change (ΔS) were derived accordingly Anderson-Freeman method interpreted in light of nature of the substituents. The nature, size and position of substituents affected thermal properties of the Schiff bases.

Return words of linear involutions and fundamental groups

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet,$A$. The set of return words with respect to a subgroup of finite index$G$of the free group on$A$is also proved to be a symmetric basis of$G$.

Solute–solvent interactions in chloroform solutions of halogenated symmetric double Schiff bases of 1,1′-bis(4-aminophenyl)cyclohexane at 308.15 K according to ultrasonic and viscosity data

The density, viscosity and ultrasonic speed (2 MHz) of chloroform solutions of halogenated symmetric double Schiff bases of 1,1′-bis(4-aminophenyl)cyclohexane were investigated at 308.15 K. Various acoustical parameters such as specific acoustical impedance (Z), adiabatic compressibility (К a), Rao’s molar sound function (R m), van der Waals constant (b), internal pressure (π), free volume (V f), intermolecular free path length (L f), classical absorption coefficient (α/f 2)Cl) and viscous relaxation time (τ) were determine using ultrasonic speed (U), viscosity (η) and density (ρ) data of Schiff bases solutions and correlated with concentration. Linear increase of Z, b, R, τ, and (α/f 2)Cl except π (nonlinear) and linear decrease of К a and L f except V f (nonlinear) with increasing concentration of Schiff bases suggested presence of strong molecular interactions in the solutions. The positive values of solvation number further supported strong molecular interactions in the solutions. The nature and position of halogen substituent also affected the strength of molecular interactions.

Cantor–Bernstein theorems for certain symmetric bases in Banach spaces

We prove the following Cantor–Bernstein type theorem, which applies well to the class of symmetric sequence spaces studied earlier by Altshuler, Casazza, and Lin: Let X and Y be Banach spaces having symmetric bases (xn) and (yn), respectively. If each of the bases (xn) and (yn) is equivalent to a basic sequence generated by one vector of the other, then the spaces X and Y are isomorphic. As a consequence, we obtain the strong equivalence that two Lorentz sequence spaces have the same linear dimension if and only if they are isomorphic.

On the computational implementation of forward and back-projection operations for cone-beam computed tomography.

Forward- and back-projection operations are the main computational burden in iterative image reconstruction in computed tomography. In addition, their implementation has to be accurate to ensure stable convergence to a high-quality image. This paper reviews and compares some of the variations in the implementation of these operations in cone-beam computed tomography. We compare four algorithms for computing the system matrix, including a distance-driven algorithm, an algorithm based on cubic basis functions, another based on spherically symmetric basis functions, and a voxel-driven algorithm. The focus of our study is on understanding how the choice of the implementation of the system matrix will influence the performance of iterative image reconstruction algorithms, including such factors as the noise strength and spatial resolution in the reconstructed image. Our experiments with simulated and real cone-beam data reveal the significance of the speed-accuracy trade-off in the implementation of the system matrix. Our results suggest that fast convergence of iterative image reconstruction methods requires accurate implementation of forward- and back-projection operations, involving a direct estimation of the convolution of the footprint of the voxel basis function with the surface of the detectors. The required accuracy decreases by increasing the resolution of the projection measurements beyond the resolution of the reconstructed image. Moreover, reconstruction of low-contrast objects needs more accurate implementation of these operations. Our results also show that, compared with regularized reconstruction methods, the behavior of iterative reconstruction algorithms that do not use a proper regularization is influenced more significantly by the implementation of the forward- and back-projection operations.

Conventional and Microwave Irradiated Syntheses of New Symmetric Double Schiff Bases of 1,1’-Bis(4-Amino Phenyl)Cyclohexane, their Spectral Characterization and Microbial Activity

Symmetric double Schiff bases of 1,1’- bis (4-amino phenyl) cyclohexane and substituted aromatic benzaldehydes were synthesized by thermal and microwave irradiated green chemistry techniques. The purity of Schiff bases was checked by TLC. The structure of Schiff bases were supported by UV, FTIR, 1HNMR and MS techniques. The antibacterial and antifungal activities of Schiff bases were checked against gram positive and gram negative microbes. Schiff bases showed moderate antibacterial activity but they showed good comparable antifungal activity against chosen microbes and standard drugs.

Conventional and Microwave Irradiated Syntheses of New Symmetric Double Schiff Bases of 1,1’-Bis(4-Amino Phenyl)Cyclohexane, their Spectral Characterization and Microbial Activity

Symmetric double Schiff bases of 1,1’- bis (4-amino phenyl) cyclohexane and substituted aromatic benzaldehydes were synthesized by thermal and microwave irradiated green chemistry techniques. The purity of Schiff bases was checked by TLC. The structure of Schiff bases were supported by UV, FTIR, 1HNMR and MS techniques. The antibacterial and antifungal activities of Schiff bases were checked against gram positive and gram negative microbes. Schiff bases showed moderate antibacterial activity but they showed good comparable antifungal activity against chosen microbes and standard drugs.

Density, viscosity and speed of sound based acoustical properties of halogenated symmetric double Schiff bases of 1,1'-bis(4-aminophenyl)cyclohexane in 1,4-dioxane at 308.15 K

Density, viscosity and speed of sound based acoustical properties of halogenated symmetric double Schiff bases of 1,1'-bis(4-aminophenyl)cyclohexane in 1,4-dioxane at 308.15 K

Microwave Irradiated and Conventional Syntheses of Halogenated Symmetric Double Schiff Bases, their Spectral and Microbial Study

The aim of the present work was to discover various synthetic route of biological active halogenated symmetric double Schiff bases of 1,1’-bis (4-aminophenyl) cyclohexane, their spectral and microbial study. Symmetric double Schiff base ligands of halogenated benzaldehyde and 1,1’-bis (4-aminophenyl) cyclohexane were synthesized by conventional and green chemistry approach. The purity of Schiff bases was checked by Thin Layer Chromatography (TLC). The structures of these ligands were elucidated by UV, FTIR, 1HNMR and MS; and assayed for their antibacterial activity against Escherichia coli (E. coli), Bacillus megaterium (B.mega), Proteus vulgaris (P. vulgaris, Staphylococcus aureus (S. aureus) and Aspergillus niger (A. niger)The minimum inhibition concentrations (MIC) of the compounds were studied by cup plate method. Schiff bases revealed moderate antimicrobial activity and good antifungal activity against chosen standard drugs and microorganisms.

Microwave Irradiated and Conventional Syntheses of Halogenated Symmetric Double Schiff Bases, their Spectral and Microbial Study

The aim of the present work was to discover various synthetic route of biological active halogenated symmetric double Schiff bases of 1,1’-bis (4-aminophenyl) cyclohexane, their spectral and microbial study. Symmetric double Schiff base ligands of halogenated benzaldehyde and 1,1’-bis (4-aminophenyl) cyclohexane were synthesized by conventional and green chemistry approach. The purity of Schiff bases was checked by Thin Layer Chromatography (TLC). The structures of these ligands were elucidated by UV, FTIR, 1HNMR and MS; and assayed for their antibacterial activity against Escherichia coli (E. coli), Bacillus megaterium (B.mega), Proteus vulgaris (P. vulgaris, Staphylococcus aureus (S. aureus) and Aspergillus niger (A. niger)The minimum inhibition concentrations (MIC) of the compounds were studied by cup plate method. Schiff bases revealed moderate antimicrobial activity and good antifungal activity against chosen standard drugs and microorganisms.

Density, Viscosity and Ultrasonic Speed Based Acoustical Studies of Structural Properties of 1,1’-Bis(4-Amino Phenyl) Cyclohexane Symmetric Double Schiff Bases in THF Solutions at 303.15, 308.15 and 313.15K

The density (ρ), viscosity (η) and ultrasonic speed (U) (2MHz) of THF solutions of symmetric double Schiff bases (SDSB-1,SDSB-2 and SDSB-3) were determined at 303.15, 308.15 and 313.15 K. Various acoustical parameters such as specific acoustical impedance (Z), isentropic compressibility (ks), Rao’s molar sound function (Rm), Van der Waals constant (b), internal pressure (π), free volume (Vf), intermolecular free path length (Lf), viscous relaxation time (τ) and classical absorption coefficient (α/f2)Cl), were determined using ρ, η and U data. The results are interpreted in terms of molecular interactions occurring in the solutions at different temperatures and concentrations. Linear increase of ρ, η U, Z, Rm, b, (α/f2)Cl and τ with increasing C, linear decrease of КS, Lf and π with increasing T, Vf increased linearly with C and T except SDSB-3 supported existence of strong molecular interactions in the solutions and confirming solvophilic nature of the Schiff bases. The structure, nature and size of the solutes and solvent, concentration and temperature affected molecular interactions.

Density, Viscosity and Ultrasonic Speed Based Acoustical Studies of Structural Properties of 1,1’-Bis(4-Amino Phenyl) Cyclohexane Symmetric Double Schiff Bases in THF Solutions at 303.15, 308.15 and 313.15K

The density (ρ), viscosity (η) and ultrasonic speed (U) (2MHz) of THF solutions of symmetric double Schiff bases (SDSB-1,SDSB-2 and SDSB-3) were determined at 303.15, 308.15 and 313.15 K. Various acoustical parameters such as specific acoustical impedance (Z), isentropic compressibility (ks), Rao’s molar sound function (Rm), Van der Waals constant (b), internal pressure (π), free volume (Vf), intermolecular free path length (Lf), viscous relaxation time (τ) and classical absorption coefficient (α/f2)Cl), were determined using ρ, η and U data. The results are interpreted in terms of molecular interactions occurring in the solutions at different temperatures and concentrations. Linear increase of ρ, η U, Z, Rm, b, (α/f2)Cl and τ with increasing C, linear decrease of КS, Lf and π with increasing T, Vf increased linearly with C and T except SDSB-3 supported existence of strong molecular interactions in the solutions and confirming solvophilic nature of the Schiff bases. The structure, nature and size of the solutes and solvent, concentration and temperature affected molecular interactions.

Synthesis, characterization, and in vitro antioxidant and anticancer studies of ruthenium(III) complexes of symmetric and asymmetric tetradentate Schiff bases

Ruthenium(III) complexes of three tetradentate Schiff bases with N2O2 donors formulated as [RuCl(LL1)(H2O)], [RuCl(LL2)(H2O)] and [RuCl(LL3)(H2O)] were synthesized and characterized by elemental analyses, molar conductance, FTIR, and electronic spectral measurements. The FTIR data showed that the tetradentate Schiff base ligands coordinate to Ru ions through the azomethine nitrogen and enolic oxygen. The antioxidant activities of the complexes were investigated through scavenging activity on 1,1-diphenyl-2-picrylhydrazyl (DPPH) and 2,2′-azinobis-3-ethylbenzothiazoline-6-sulfonic acid (ABTS) radicals. The DPPH activity for [RuCl(LL2)(H2O)] with IC50 = 0.031 mg mL−1 was higher than the values obtained for the other Ru(III) compounds. The study revealed that the synthesized Ru(III) complexes of the tetradentate Schiff base exhibited strong scavenging activities against DPPH and moderate against ABTS radicals. In addition, the antiproliferative studies of the complexes were also tested against human renal cancer cells (TK10), human melanoma cancer cells (UACC62), and human breast cancer cells (MCF7) using the SRB assay. The results indicated that the Ru(III) complexes showed low anticancer activities against the tested human cancer cell lines.

Basis Functions in Image Reconstruction From Projections: A Tutorial Introduction

The series expansion approaches to image reconstruction from projections assume that the object to be reconstructed can be represented as a linear combination of fixed basis functions and the task of the reconstruction algorithm is to estimate the coefficients in such a linear combination based on the measured projection data. It is demonstrated that using spherically symmetric basis functions (blobs), instead of ones based on the more traditional pixels, yields superior reconstructions of medically relevant objects. The demonstration uses simulated computerized tomography projection data of head cross-sections and the series expansion method ART for the reconstruction. In addition to showing the results of one anecdotal example, the relative efficacy of using pixel and blob basis functions in image reconstruction from projections is also evaluated using a statistical hypothesis testing based task oriented comparison methodology. The superiority of the efficacy of blob basis functions over that of pixel basis function is found to be statistically significant.