In this paper, we show that the Relative algebraic Interior is a suitable replacement for both of the topological Interior and the algebraic Interior for the cases where these are empty. Also, we present some properties of (Relative) algebraic Interior and some fixed point theorems for increasing mapping. The obtained results can be viewed as an extension and improvement of the known corresponding results. Also, some examples are to support our conclusions considered.

This paper proposes a new linear program for finding a Relative Interior point of a polyhedral set. Based on characterizing the Relative Interior of a polyhedral set through its polyhedral representing sets, two main contributions are made. First, we complete the existing results in the literature that require the non-negativity of the given polyhedral set. Then, we deal with the general case where this requirement may not be met.

One of the fundamental concepts in convex analysis and optimization is the Relative Interior of a set. This concept is used when the Interior of a set is empty due to the incompleteness of its dimension. In this paper, first, we propose a linear programming model to find a Relative Interior point of a polyhedral set. Then, we discuss the application of this model to geometric programming. Specifically, we show that a special form of our proposed model can determine the degeneracy of a geometric programming problem by identifying a Relative Interior point of the feasible region of its dual. Finally, we present two numerical examples for describing the applications of the proposed model.

Thirty samples were examined to document the nannofossil assemblage and determine the biostratigraphic zonation. Samples were taken from a borehole drilled at Sisseton (NE South Dakota). 92 species of nannofossil were determined and photographed. Biostratigraphic study of calcareous nannofossils allowed recognition of Late Early Campanian to Early Late Campanian age for the studied section.

The concept of a Γ-semihyperring is a generalization of a semiring, semihyperring, Γ-semiring. In this Paper we introduce the notion of bi-Interior Γ-hyperideals, quasi-Interior Γ-hyperideals and bi-quasi-Interior Γ-hyperideals in a Γ-semihyperring as a generalization of Γ-hyperideal, left-Γ-hyperideal, right-Γ-hyperideals, bi Γ-hyperideal, quasi Γ-hyperideal, Interior Γ-hyperideals of Γ-semihyperring. We studied the properties of these Γ-hyperideals and characterized them in simple Γ-semihyperring and regular Γ-semihyperring

"Interior decoration", "Interior design", and "Interior architecture" are known as three names that refer to specialties that are focused on the organization of Interior spaces. The emergence of these three specialties takes place in a historical process; first "Interior decoration" in the early nineteenth century, then "Interior design" in the early twentieth century, and then "Interior architecture" about four decades ago. The precedence and delay in the emergence and application of these specialties does not mean the complete replacement of the latter over the earlier one; however, in some cases and for some reasons, the former specialty has been replaced by the latter. In fact, these three specialties, despite their many similarities and commonalities, have different meanings and uses and in relation to each other, and they have created a relationship that is questionable and worth considering. So the main question of this research is what is the relationship between the mergence of "Interior decoration", "Interior design", and "Interior architecture".To answer the above question, first, the foundations and origins of each are examined and analyzed. This is because by finding the roots and recognizing the factors and conditions of each period, it is possible to identify the requirements and needs that have affected the emergence of each. These factors and conditions form a system in which they interact and are affected. As a result, their interaction leads to the formation and objectification of each of these specialties. After recognizing the way the three specialties are formed, their function and functional evolution have been studied in both the context of profession and education. In this way, the relationship between these three specialties is explained. Since the subject of this research is to know the origin of these professional areas and how it is formed, the main focus is on beginning to find the subject matter of these specialties that have lasted from the early nineteenth century to the late twentieth century. In the last two decades, these specialties have become more widespread in the profession and education, and so a careful study of this issue requires further opportunities. Therefore, the time period studied in this research is from around 1800 to 2000.So the goal of this study is to understand the relationship between these three titles. The methodology of this research is analytical, interpretive, and logical reasoning. This means that by searching, collecting, and organizing documents, and evaluating and interpreting data, a description of the origin and formation of these three specialties is provided. The findings of this study show how the factors influencing the formation and survival of "Interior decoration", "Interior design", and "Interior architecture" operate and how they relate to each other. Since a Relatively short time has passed since the introduction of these specialties, this research can help to understand each of these three areas of expertise more accurately and is effective in clarifying their position in relation to each other. Continuation of this research can help to develop the concepts and theoretical foundations of these specialties.

Due to the importance of Tabriz in various historical periods after the Mongol’s invasion to the end of the Qajar period, as continuously being the capital city or the residence city of the prince, entrance to this city has been reflected in various reports. Most of the foreigners who entered Iran from European countries came from Caucasus Road to Tabriz and reported many buildings on their way. Moreover, this road was one of the most significant roadways for Caravans on the Silk Road, where various residential buildings have been constructed. Alaki caravansarai is one of the most important roadside accommodations in Northwest Iran, and so many reports about it and the quality of its façade decorations can be traced in the travelogues of foreigners who traveled to Iran. Its vastitude and elegant entrance gate nevertheless, this building has unfortunately been neglected and was about to be ruined in the 19th century and afterward. However, the beauty of its entrance gate has gained the attention of many people and has been reported in many sources. Based on the ornaments used, recent sources in the history of architecture have estimated its date to the 12th century and Ilkhanid Abū Saʿīd Bahādor Khan and the Timurid period. These sources based their estimation only on the type of ornaments used on its entrance gate and its probable that the existence of dated artworks of the Ilkhanid period nearby, such as the Mihrab of Marand’s Jame Mosque, has influenced this speculation. This article aims to meticulously study this building in various travelogues, al-Wqafīyya al-Rashīdīyya, and other sources in the history of architecture and discusses the ornament types used and their history. The detailed study of the documents and sources indicated that this building could belong to the presidency period of Ḵᵛāja Rašid-al-Din Fażl-Allāh, and based on his emphasis to establish this building, can be dated 1297-1309. The ornaments, moreover, indicate the transient period from brickwork to tilework which happened in the late 13th and early 14th centuries

Identifying the constraining factors of production and yield gap is very important. Therefore; this research was performed to identify the production constraining factors of local rice cultivars. All management practices from nursery preparation to harvesting stages for 100 paddy fields of local rice cultivars were recorded through field studies, in Sari, from 2015-2016. In the CPA, the actual and calculated potential yield were 4495 and 5703 kg/ha, respectively and the gap was 1221 kg/ha. The yield gap caused by number of top-dressing variables was 324 kg/ha, equal to 27% of the total yield gap. The yield gap related to previous year of legumes cultivation was 218 kg ha-1, equal to 18% of the total yield variation. Among the 10 variables entered in the CPA model, the effects of top-dress fertilizer application and its application frequency and foliar application were remarkable, which could compensate a significant part of the yield gap (444 kg/ha, 37% of total) in the farmers’ fields by managing these variables. According to boundary line analysis (BLA) finding, actual yield mean on the basis of optimal limit related to 12 variables under study was 5369 kg/ha, with 881 kg/ha yield gap . Mean Relative yield and Relative yield gap for 12 variables (transplanting date, seedling age, number of seedlings per hill, planting density, nitrogen and phosphorous per hectare, nitrogen before transplanting, harvesting date, lodging problem, pest problem, diseases problem and weeds problem) were 83.64 and 16.35 kg/ha, respectively. Based on the finding, it can be stated that the model precision is appropriate and can be applied for both estimation of the quantity of yield gap and determining the portion of each restricting yield variables.

Hesitant fuzzy sets are very useful for dealing with group decision-making problems when experts have hesitation among several possible memberships for an element in a set. Recently, many researchers have used these concepts in clustering analysis and decision-making. In this article, the notions of hesitant hybrid subsemigroups and hesitant hybrid left(resp., right) ideals in a semigroup are introduced, and several properties are investigated. The concept of hesitant hybrid product is also introduced, and properties of hesitant hybrid subsemigroups and hybrid left (resp., right) ideals are considered using the notion of hesitant hybrid product. Relations between hesitant hybrid intersection and hesitant hybrid product are exhibited. Additionally, we establish hesitant hybrid Interior ideals in semigroups and study theirproperties. Furthermore, we prove that in regular and intra-regular semigroups, the hesitant hybrid ideal and the hesitant hybrid Interior ideals are coincide.