This paper focuses on investigating the equivalence problem for fifth-order differential operators (FODOs) on the line under general fiber-Preserving transformations. Utilizing the Cartan method of equivalence, the study specifically addresses the gauge equivalence problem, seeking to establish the conditions for two FODOs to be related by a fiber-Preserving transformation. By analyzing the properties of these operators, the research aims to identify conditions for their transformation while maintaining the fiber structure. The systematic approach of the Cartan method is employed to derive the necessary conditions for gauge equivalence between these FODOs. The study aims to enhance understanding of the equivalence problem for FODOs and shed light on fiber-Preserving transformations that uphold gauge equivalence.

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. Let ϕ, : A →,B be a bijective map. In this paper, we show that ϕ,is completely Preserving quadratic operator in both directions if and only if ϕ,is 2-quadratic Preserving operator in both directions and if and only if ϕ,is either an isomorphism or (in the complex case) a conjugate isomorphism.

The problem of personal identity among others may stem from the following question—what does be the person that you are, from one day to the next, necessarily consist of? The diachronic problem of personal identity raises question on the necessary and sufficient conditions for the identity of the person over time. The synchronic problem is grounded in the question of what features or traits characterize a given person at one time. To answer these questions, John Locke discarded the soul and the body as necessary and sufficient substances for personal identity over time. He accepted consciousness as the only criterion for personal identity; the only thing capable of remaining the same and Preserving personal identity through change. Though Locke’s argument is somewhat clear and coherent but what remains vague and incoherent is embedded in the question—what exactly is consciousness? How and why should it be the basis or criterion for the determination of personal identity? Using the method of critical analysis, I argue that Locke’s choice of consciousness as the determinant of personal identity, though quite novel, is incoherent and vague. Secondly, Locke had already presumed and anticipated clearly though fallaciously the very thing he wishes to substantiate. I therefore conclude that Locke’s argument is just another way of trying to escape but inadvertently prolonging the difficulty of apparently articulating a distinction between the psychological approach and physiological approach to the problem of personal identity. However, in my submission, I propose the concept of the “other” as alternative approach— a sort of an extrinsic-intrinsic approach to the problem of personal identity.

In Kant and the problem of Metaphysics (KPM), Heidegger shows how the initial image-making power of the imagination is pictorial in the pure image of time. He further clarifies the recognized link between the syntheses and the time-visualizing power of images which is pictorial formative power, which can visualize time. The three syntheses of imagination are replaced with the modes of existence of Dasein. Three kinds of photographs resemble the images produced when the transcendental imagination functions as Dasein. We can address the ways to visualize time by considering Dasein with a photographic preoccupation through 1. A possible look in the form of presentation (a photograph shows an immediate look of a house or person as intuitable this-here); 2. An immediate look of the thing, house, or deceased person, as well as a copy or its reproduction as a death mask; and, 3. The manner of contemplation of a likeness is an immediate look of a being present in the form of representation.

Considering the importance of solidarity as a theoretical and political structure and its connection with social harmony, acceptability and political legitimacy, the present research has conceptualized and formulated the problem of solidarity in the context of Iran's multi-ethnic society through the mediation of existing studies. . In the current research, the hybrid method has been used. The sample and statistical population of the research includes selected qualitative studies that have investigated the issue of solidarity and its dimensions in Iran from different aspects and perspectives in the period from 1372 to 1398. After searching in different databases, 500 related articles, books and research reports were identified and finally 54 sources were selected to perform the meta-combination method. Based on the synthesis of studies, solidarity in the multi-ethnic society of Iran has found a problematic situation. The results of the research confirm that the coordinates of solidarity in Iranian society are based on structural contexts and conditions. The main field in which the issue of solidarity in Iran should be established is the acceptance of a reality called "reliability and stability of cultural diversity". Based on the meta-combination of studies, the coordinates of solidarity in Iran's multi-ethnic society, due to "intercultural affinities", "disruption and non-inclusion in ethnic policies", "instability and imbalance of development", "inefficiency of ethnic political management" and "identification disorder". In the meantime, "cultural globalization", "taking role models from ethnic elites", "geopolitics of ethnic areas and transnational interventions" and "ethnic extension" are considered as intervening conditions. Despite the fact that ethnic and religious diversity is recognized as a central sign in the constitution, with this definition, political agents have sometimes resorted to strategies that social suffering and extroversion has formed in ethnic communities.

The target of this paper is to present partial fuzzy metric-Preserving functions and characterize the functions $f:[0,1]\to[0,1]$ with this aspect. We give a characterization for partial fuzzy  metric-Preserving functions considering the different t-norms. Also, we show that the topology induced by partial fuzzy metric does not preserve under these functions with  an example. Then we  give a characterization of those partial fuzzy metric-Preserving functions which preserve completeness and contractivity under some conditions. Finally, we discussed the relation between fuzzy  metric Preserving and partial fuzzy Preserving functions.