Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrödinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limit0.     

Bounds on Exponential Sums and the Polynomial Waring Problem Mod p

Estimates are given for the exponential a prime and f a nonzero integer polynomial, of interestin cases where the Weil bound is worse than trivial. The resultsextend those of Konyagin for monomials to a general polynomial.Such bounds readily yield estimates for the corresponding polynomialWaring problem mod p, namely the smallest such that f(x₁)+...+f(x)N(mod p) is solvable in integers for any N.     

Cytotoxicity and hydrolysis of trans-Ti(IV) complexes of salen ligands: structure-activity relationship studies

Eleven bis(dimethylphenolato) Ti(IV) complexes of salen ligands with different steric and electronic properties due to different aromatic substituents at the ortho and para positions are reported, and their cytotoxicity toward HT-29 and OVCAR-1 cells and its dependence on hydrolytic behavior are discussed. Eight complexes of this series were analyzed by X-ray crystallography, confirming the trans geometry of the labile ligands with otherwise relatively similar coordination features to those of cis-salan analogues. Relatively high and similar hydrolytic stability is observed for all complexes, with t(1/2) values for labile ligand hydrolysis of 2-11 h in 10% D(2)O solutions. In contrast, varying cytotoxicities were achieved, identifying selected members as the first trans-Ti(IV) complexes reported as anticancer agents. Steric bulk all around the complex diminished the activity, where a complex with no aromatic substitutions is especially active and complexes substituted particularly at the ortho positions are mostly inactive, including ortho-halogenated and ortho-tert-butylated, with one exception of the ortho-methoxylated complex demonstrating appreciable activity. In contrast, para-halogenation provided the complexes of highest cytotoxic activity in this series (IC(50) as low as 1.0 ± 0.3 μM), with activity exceeding that of cisplatin by up to 15-fold. Reaction of a representative complex with ortho-catechol yielded a "cis"-Ti(IV) complex following rearrangement of the salen ligand on the metal center, with highly similar coordination features and geometry to those of the catecholato salan analogues, suggesting that the complexes operate by similar mechanisms and rearrangement of the salen ligand may occur upon introduction of a suitable chelating target. In additional cytotoxicity measurements, a salen complex was preincubated in the biological medium for varying periods prior to cell addition, revealing that marked cytotoxicity of the salen complex is retained for longer preincubation periods relative to known Ti(IV) complexes, suggesting that the hydrolysis products may also induce cytotoxic effects, thus reducing stability concerns.     

Trans titanium(IV) complexes of salen ligands exhibit high antitumor activity

Salen-titanium(IV) complexes are introduced as a new family of highly efficient antitumor complexes, being the first cytotoxic titanium(IV) complexes of trans labile ligands, as characterized crystallographically. Four complexes with different aromatic substitutions were analyzed, reveling a meaningful effect of the ligand structure on the complex performance. All complexes exhibit high hydrolytic stability, where the labile OAr ligands hydrolyze in a 10% D(2)O solution with t(1/2) ranging from 2 to 11 h. The IC(50) values obtained for three of the salen complexes studied on HT-29 colon and OVCAR-1 ovarian cells demonstrate activity that exceeds those of the known tianium(IV) complexes Cp(2)TiCl(2) and (bzac)(2)Ti(OiPr)(2) and that of cisplatin, where the most active para-chlorinated complex exhibits activity enhancement relative to cisplatin by 10-fold.     

3-D Model of sound pressure field in a meridinal section plane of fruit

A theoretical model was suggested for qualitative evaluation of a sound pressure field in fruit tissue, as affected by ultrasonic probe dimensions and fruit properties. The classic directivity pattern of an ideal fluid model, expressed by Bessel function of the first kind, was extended to include energy dissipation of a real material. The directional characteristics of wave propagation, as influenced by transmitter frequency and diameter, and by fruit properties, were discussed. The model indicates how to select the parameters of the ultrasonic transducer (transducer diameter, frequency and excitation power) to control the magnitude and directivity of the ultrasonic waves in the fruit tissue. The suggested theoretical model represented fairly well the experimental sound wave distribution over the half-cut surface of potato and avocado (R² > 0.862 and 0.977, respectively); the same theoretical model could not represent the sound wave distribution over a half-cut melon. Results of the study were applied in a new probe design for ultrasonic testing of whole fruit.     

Use of lagrange multipliers with polynomial series for dynamic analysis of constrained plates part I: Polynomial series

A theory for prediction of the dynamic response of a constrained plate is presented here. The boundaries of the plate may be partially fixed, its dynamic loading is due to elastically mounted vibrating machines and its constraints include beam-like stiffeners. The theory yields the eigenvalues and modal shapes of the plate and stiffeners which comprise the system. The solution, given in Part I, is based on Galerkin's method combined with use of special polynomial series presented by Kantorovich and Krylov. These eigenvalues are used in Part II [1] for response analysis of the complete system and the eigenvalues of the complete system will be obtained by the application of Lagrange equations and multipliers. The various coefficients used in the process are presented in the Appendices to the work. Comparisons with published results show good agreement.     

Acoustic wave propagation in bent thin-walled wave guides

Theoretical analysis of sound wave propagation in structures indicates considerable amplitude reduction during transmission through a bent joint, while some build-up can be observed in front of the bend. Every type of transmitted wave (longitudinal and flexural in plane frames) is found to combine with other forms of sound propagation as a result of the deflection of the axis of the wave guide. The screening effect of the bend, in solids with arbitrary viscoelastic properties, is evident even under highly simplified assumptions, such as compatibility and equilibrium. By solving several practical problems, the attenuation due to folding of the structure is compared to the damping effect of the material, with the conclusion that the former contribution is the larger of the two. A pair of examples of insulation analysis of structures is worked out, and the results yield the particle velocity—the first step in determining the radiation pattern. The influence of dense columns in walls, of beams in plates, and of coupling effects in bent structures are examined.     

Theoretical models for investigation of sound transmission through isolation layers in staircase systems

The behaviour patterns of staircase dynamic response to structure-borne sound are given through analysis of two-dimensional models. Typical models of this kind include angular discontinuities which represent the intersection line between the horizontal slabs and the stair's slope. Numerical examples illustrate the effect of elastic isolation layers on the solid-borne transmission loss between the stairs and the other parts of the building, the strong coupling between the stairs and the building caused by substitution of the elastic isolation layers by rigid sound bridges, the dependence of the dynamic response of the staircase on frequency, and the coupling between the longitudinal and flexural waves due to the aforementioned angular discontinuities of the staircase system.     

A modified formula for absorption influence on sound transmission through partitions

A discrepancy between formulae for room acoustics, published in books, journals and national and international regulations, led the author to a compatible definition of ‘modified absorption area’. Using this definition consistently for various measurements of room acoustics removed the fault.A numerical example clarifies the use of the formulae and consolidates the theoretical background behind them.