Nested perfect arrays Becher, Veronica; Carton, Olivier
IEEE transactions on information theory,
08/2024
Journal Article
Recenzirano
We introduce two-dimensional periodic arrays that are a variant of the de Bruijn tori. We call them nested perfect arrays. Instead of asking that every array of a given size has exactly one ...occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for n being a power of 2, we partition the positions of the arrays in n 2 many congruence classes by taking the row number modulo n and the column number modulo n . We construct arrays where each possible n × n array occurs n 2 times, once in each congruence class. Our method yields exponentially many (in n 2 ) different nested perfect arrays.
Emergence of Internet of Things (IoT) and modern digital applications such as digital financial services and deliveries make it easy to reproduce and re-distribute digital contents thus give room to ...so many copyright violations of illegal use of contents that need to be resolved. Researcher have been presenting the watermarking algorithms to prevent these illicit activities to a document before distribution. However, several issues have been identified for the digital transactions in the IoT. Thus, this research proposes a new text document image watermarking algorithm which emphasizes on two most important measures, visual quality, and robustness. To boost these measures, third least significant bit has been used for insertion. In addition, to further strengthen the technique, the Pascal Triangle is applied to determine the best position for embedding. Experimental results on the standard dataset have revealed that the proposed watermarking has achieved very encouraging results with PSNR and NCC averaged 54.95db and 0.98, respectively.
Hyperbolic Pascal triangles Belbachir, Hacène; Németh, László; Szalay, László
Applied mathematics and computation,
01/2016, Letnik:
273
Journal Article
Recenzirano
In this paper, we introduce a new generalization of Pascal’s triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the ...hyperbolic plane. We precisely describe the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearance of some binary recurrences in a fixed hyperbolic triangle are investigated.
A protein Pascal triangle has been constructed as new type of supramolecular architecture by using the inducing ligand strategy that we previously developed for protein assemblies. Although ...mathematical studies on this famous geometry have a long history, no work on such Pascal triangles fabricated from native proteins has been reported so far due to their structural complexity. In this work, by carefully tuning the specific interactions between the native protein building block WGA and the inducing ligand R‐SL, a 2D Pascal‐triangle lattice with three types of triangular voids has been assembled. Moreover, a 3D crystal structure was obtained based on the 2D Pascal triangles. The distinctive carbohydrate binding sites of WGA and the intralayer as well as interlayer dimerization of RhB was the key to facilitate nanofabrication in solution. This strategy may be applied to prepare and explore various sophisticated assemblies based on native proteins.
A Pascal‐triangle lattice is constructed by using a careful selection of protein building blocks with anisotropic shapes and two sets of carbohydrate binding sites. Moreover, the dynamic and exchangeable nature of non‐covalent interactions makes the further manipulation of 2D lattices to 3D crystals possible.
Power sums in hyperbolic Pascal triangles Németh, László; Szalay, László
Analele Universităţii "Ovidius" Constanţa. Seria Matematică,
3/2018, Letnik:
26, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to –4, q} with q ≥ 5. The method is based on the theory of ...linear recurrences, and the results are demonstrated by evaluating the k
power sum in the range 2 ≤ k ≤ 11.
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