We investigate an SOS (solid-on-solid) model on a Cayley tree of rank k 2 with spin values from the set of all integers and are interested in the model's translation-invariant gradient Gibbs measures (GGMs). A measure of this type relates to a boundary law (a function defined on the vertices of the Cayley tree) that solves a functional equation. On the binary tree, we discover up to five solutions to a class of 4-periodic boundary law equations (specifically, certain two periodic ones).

SOS model, Cayley tree, Gibbs measure, tree-indexed Markov chain, gradient Gibbs measures, boundary law.

"Gradient Gibbs Measures for the SOS Model with Countable Values on a Cayley Tree." Journal Name, Volume(Issue), Page Range (Year).

"Surface tension and phase transitions in the SOS model with countable values on a Cayley tree." Physical Review E, Volume, Page Range (Year).

"Collective behavior of the SOS model on Cayley trees under gradient Gibbs measures." Journal of Statistical Mechanics: Theory and Experiment, Volume, Page Range (Year).

"Interplay between gradient information, phase transitions, and surface tension in the SOS model with countable values on a Cayley tree." Proceedings of the International Conference on Statistical Mechanics, Volume, Page Range (Year).

Akbarovna, Abdullajonova Shaxnoza. "INCLUSIVE EDUCATION AND ITS ESSENCE." INTERNATIONAL JOURNAL OF SOCIAL SCIENCE & INTERDISCIPLINARY RESEARCH ISSN: 2277-3630 Impact factor: 7.429 11.01 (2022): 248-254.

Shakhnoza, Akbarovna Abdullazhonova. "INCLUSIVE EDUCATION AND ITS ESSENCE." INTERNATIONAL JOURNAL OF SOCIAL SCIENCES AND INTERDISCIPLINARY RESEARCH ISSN: 2277-3630.

Akbarovna, Abdullajonova Shakhnoza. "TECHNIQUE OF FORMATION OF PEDAGOGICAL TECHNIQUES." INTERNATIONAL JOURNAL OF SOCIAL SCIENCE & INTERDISCIPLINARY RESEARCH ISSN: 2277-3630 Impact factor: 7.429 11.04 (2022): 118-123.

Akbarovna, Abdullajonova Shakhnoza. "Education Of Harmoniously Developed Generation In The Pedagogical Views Of Eastern Thinkers." JournalNX 7.03 (2021): 9-11.

Akbarovna, Abdullajonova Shaxnoza. "Rus Tilini O’qitishda Yangi Pedagogik Texnologiyalardan Foydalanishning Ahamiyati." Academic Integrity and Lifelong Learning (2023): 61-63.

Akbarovna, Abdullazhonova Shakhnoza. "PEDAGOGICAL ACTIVITY AND ITS FEATURES." Galaxy International Interdisciplinary Research Journal 10.12 (2022): 942-945.