Special Diophantine Triples Involving Square Pyramidal Numbers

In this communication, we accomplish special Diophantine triples comprising of square pyramidal numbers such that the product of any two members of the set added by their sum and increased by a polynomial with integer coefficient is a perfect square.

Special Diophantine Triples InvolvingSquare Pyramidal Numbers

In this communication, we accomplish special Diophantine triples comprising of square pyramidal numbers such that the product of any two members of the set added by their sum and increased by a polynomial with integer coefficient is a perfect square.

Image Watermarking Based on IWT and Parity Bit Checking

Various document types play an influential role in a lot of our lives activities today; hence preserving their integrity is an important matter. Such documents have various forms, including texts, videos, sounds, and images. The latter types' authentication will be our concern here in this paper. Images can be handled spatially by doing the proper modification directly on their pixel values or spectrally through conducting some adjustments to some of the addressed coefficients. Due to spectral (frequency) domain flexibility in handling data, the domain coefficients are utilized for the watermark embedding purpose. The integer wavelet transform (IWT), which is a wavelet transform based on the lifting scheme, is adopted in this paper in order to provide a direct way for converting image pixels' integer values to integer coefficient values rather than floating point coefficients that could be produced by the traditional wavelet transform. This direct relation can enhance the processed image quality due to avoiding the rounding operations on the floating point coefficients. The well-known parity bit approach is also utilized in this paper as an authentication mechanism, where 3 secret parity bits are used for each block in an image which is divided into non-overlapped blocks in order to enforce a form of fragile watermark approach. Thus, any alteration in the block pixels could cause the adopted (even) parity to be violated. The fragile watermarking is achieved through the modification of least significant bits ((LSBs) of certain frequency coefficients' according to the even parity condition. In spite of this image watermarking operation, the proposed method is efficient. In order to prove the efficiency of our proposed method, it was tested against standard images using measurements like peak signal to noise ratio (PSNR) and structural similarity index (SSIM). Experiments showed promising results; the method preserves high image quality (PSNR≈ 44.4367dB, SSIM≈ 0.9956) and good tamper detection capability.

IMAGE RESOLUTION ENHANCEMENT OF SATELLITE IMAGES USING DYADIC INTEGER COEFFICIENT WAVELET FILTER FOR WAVELET TRANSFORM

In this paper, image resolution enhancements for satellite images are proposed using dyadic integer coefficients based wavelet filter (DICWF). We proposes a technique in which discrete wavelet transform and stationary wavelet transform using DICWF which is used to obtain a high resolution image and this image is derived from frequency subbands. The satellite images play a very vital role now days in the development of technical aspects which needs to be enhanced. These satellite images are superresolved with the help of dyadic integer coefficient-based wavelet filters, which reduces the hardware complexity and computational difficulties due to the rational and integer coefficients of these filter banks. The value of the peak signal-to-noise ratio (PSNR) of the proposed method and the resultant visual images of the proposed method show the effectiveness of this algorithm over other existing algorithms using discrete wavelet transform. Noise can be minimized by applying thresholding on different frequency subbands which obtained by the application of DICWF to the noisy, blurred input images.

Symplectic quandles and parabolic representations of 2-bridge knots and links

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

Divide and Conquer: Towards Faster Pseudo-Boolean Solving

The last 20 years have seen dramatic improvements in the performance of algorithms for Boolean satisfiability---so-called SAT solvers---and today conflict-driven clause learning (CDCL) solvers are routinely used in a wide range of application areas. One serious short-coming of CDCL, however, is that the underlying method of reasoning is quite weak. A tantalizing solution is to instead use stronger pseudo-Boolean (PB) reasoning, but so far the promise of exponential gains in performance has failed to materialize---the increased theoretical strength seems hard to harness algorithmically, and in many applications CDCL-based methods are still superior. We propose a modified approach to pseudo-Boolean solving based on division instead of the saturation rule used in [Chai and Kuehlmann '05] and other PB solvers. In addition to resulting in a stronger conflict analysis, this also improves performance by keeping integer coefficient sizes down, and yields a very competitive solver as shown by the results in the Pseudo-Boolean Competitions 2015 and 2016.