Product Of Sine Research Articles

Image Compression of 2-D Continuous Exponential Functions, Continuous Periodic Functions and Product of Sine and Cosine Functions using Discrete Wavelet Transform

Wavelet transforms plays an important role in image compression techniques that developed recently. Here three 2-D functions are considered which are approximated and compressed using multilevel discrete 2-D wavelet transforms like Haar, Daubechies, Coiflet and Symlet. The images that are compressed are tested for quality using the error metrics like mean square error (MSE), peak to signal noise ratio (PSNR), maximum error (MAXERR), L2RAT, compression ratio and bit per pixel. The evaluation of the above mentioned wavelets is synthesized in terms of experimental results which demonstrates that Haar wavelets provides high compression ratios for 2-D exponential functions and the product of sine and cosine functions whereas Daubechies wavelet gives good compression ratio for 2-D periodic function.

Generalized shear of a soft rectangular block

The problem of the simple shear of a block has been treated in terms of a shear displacement, applied uniformly in a lateral direction and assumed to be a linear function of the height above the base. In this paper, simple shear is generalized: the shear displacement is neither uniform in the lateral direction nor necessarily a linear function of the height. Using second-order isotropic elasticity, the analytical solutions show that the shear displacements are characterized by the product of sine and hyperbolic sine functions of the height and depth variables, respectively. The height dependence of the shear displacement is predicted to be a combination of linear and sinusoidal functions, and is verified against the test data of agar–gelatin cuboidal blocks. If the gravity effect is incorporated, a quadratic dependence on height is additionally predicted. The calculation of stresses reveals the presence of not only negative normal stresses but also sinusoidally varying shear stresses on the lateral planes tending to distort the block about the height direction. These results can be of great importance in tissue/cell mechanics.

Disentangling a Triangle

This observation—which seems to be absent in all the presentations of trigonometryknown to us—proves to be a convenient tool for bringing order into the garden oftrigonometric identities. As an example, see Figure 2 for ways to visualize products ofsines and of cosines; they would perhaps please Napier in his experiments with suchproducts that eventually led him to the concept of logarithm.

84.47 A product of sines – once again

84.47 A product of sines – once again

84.46 A product of sines revisited

84.46 A product of sines revisited

On a product of sines

On a product of sines