We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any structure definable in.

An o-minimal expansion of an ordered group is called semi-bounded if there is no definable bijection between a bounded and an unbounded interval in it (equivalently, it is an expansion of the group by bounded predicates and group automorphisms). It is shown that every such structure has an elementary extension.

We consider reducts of the structure $\mathscr{R} = \langle\mathbb{R}, +, \cdot, <\rangle$ and other real closed fields. We compete the proof that there exists a unique reduct between $\langle\mathbb{R}, +, <, \lambda_a\rangle_{a\in\mathbb{R}}$ and R, and we demonstrate how to recover the definition of multiplication in more general contexts than the semialgebraic one. We then conclude a similar result for reducts between $\langle\mathbb{R}, \cdot, <\rangle$ and R and for general real closed fields.

A structure (M, $ ,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal (...) ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1. (shrink)

§1. Introduction. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ where algebra alone determines the ordering and hence the topology of the field:In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but (...) this will be too coarse to give a diferentiable structure.A celebrated example of how partial algebraic and topological data determines a differentiable structure is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason.The main result which we discuss here is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨M, <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of Mn, n = 1, 2, …, in some first order expansion ℳ of ⟨M, <⟩. (shrink)

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic (...) on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

We give a complete characterization of the class of upward monotone generalized quantifiers Q1 and Q2 over countable domains that satisfy the scheme Q1 x Q2 y φ → Q2 y Q1 x φ. This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff Q1 is ∃ or Q2 is ∀ (excluding trivial cases). Our result shows that in infinite domains, there are more general types of quantifiers that support these entailments.

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space \ \) is definably homeomorphic to an affine definable space with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

We formulate an analogue of Zilber's conjecture for o-minimal structures in general, and then prove it for a class of o-minimal structures over the reals. We conclude in particular that if is an ordered reduct of ,<,+,·,ex whose theory T does not have the CF property then, given any model of T, a real closed field is definable on a subinterval of.

We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals. We discuss the general open question of elimination of imaginaries in an o-minimal structure.

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G\X) < dim G for some definable ${X \subseteq G}$ then X contains a torsion point of G. Along the way we develop a general theory for the so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.

We define the notion of Euler characteristic for definable quotients in an arbitrary o-minimal structure and prove some fundamental properties.

Let M be an arbitrary structure. Then we say that an M -formula φ defines a stable set inM if every formula φ ∧ α is stable. We prove: If G is an M -definable group and every definable stable subset of G has U -rank at most n , then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in an o-minimal structure.More generally, (...) an M -definable set X is weakly stable if the M -induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (shrink)

Let $\mathcal{M}$ be an o-minimal expansion of a real closed field $R$. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of $\langle R^{n},+\rangle$ contains a definable generic set and therefore admits a lattice.

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

Given a weakly o-minimal structure${\cal M}$and its o-minimal completion$\bar{{\cal M}}$, we first associate to$\bar{{\cal M}}$a canonical language and then prove thatTh$\left$determines$Th\left$. We then investigate the theory of the pair$\left$in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every definable open subset of${\bar{M}^n}$is already definable in$\bar{{\cal M}}$.We give an example of a weakly o-minimal structure interpreting$\bar{{\cal M}}$and show that it is not elementarily equivalent to any reduct (...) of an o-minimal trace. (shrink)

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. 18 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds (...) when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text]. (shrink)

The aging of the two brothers in the “twins paradox” is analyzed through the space-time evolution of the densities that correspond to their internal complex structure. Taking into account their relative motion, it is shown that the traveling brother evolves over a shorter interval of time than his twin, which makes him younger than his brother.

We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest type-definable subgroup G00 of bounded index and G/G00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

Volunteering in Ukraine is becoming an increasingly popular phenomenon that occupies an important place in the life of society. Many people choose volunteering as a way to help people in difficult life circumstances, as well as to develop their own personality and engage in active civic participation. As a significant social phenomenon, volunteering has its own ideological foundations that define its core values and principles. Volunteering is based on the desire to help people and nature regardless of their status, nationality, (...) religion, or political beliefs. The purpose of the study. The current study examines the ideological foundations of volunteering in Ukraine in wartime and their significance for volunteering, to study the practical experience of volunteers in Ukraine in 2014-2023, and to examine the peculiarities of the legal regulation of volunteering in Ukraine in wartime. Methods. Achieving the goal and objectives of the study was made possible by using a set of interrelated modern general scientific and special methods and approaches. The systematic approach made it possible to study volunteering as a holistic set of elements in the totality of the connections between them. Through the use of the method of systematization, the author provides a review of the legal framework for volunteering, factor analysis is used to determine the factors influencing the relationship between the state and society in the field of volunteering, and system analysis is used to assess the state of relations between the state and society in Ukraine at this stage through its quantitative indicators.Results. According to the study “Volunteer Movement in Ukraine”, prepared by GfK Ukraine on behalf of the UN in Ukraine, Ukrainian citizens pay great attention to the volunteer movement in the development of social processes: 62% recognize the role of volunteers in the changes in the last year; 85% believe that the volunteer movement helps strengthen peace; 81% tend to consider the volunteer movement an essential component of civil society. Regarding the legal regulation of volunteer activities in Ukraine, the Law of Ukraine “On Volunteer Activity” of April 19, 2011, No. 3236-VI was adopted in April 2011. The Law of Ukraine of 15.08.2022 No. 2519-IX On Amendments to the Law of Ukraine “On Volunteer Activity” on Support of Volunteer Activity regulates social relations related to volunteer activity in Ukraine. Discussion. One of the key principles of volunteering in Ukraine is selflessness and the ability to work in extreme conditions. Volunteers often work in difficult and dangerous conditions that require a high level of professionalism and skills. For many volunteers, the Ukrainian revolution of 2013-2014 and the events in eastern Ukraine were turning points that forced them to change their attitude to their own lives and start helping others. Ukrainian volunteers are an example of community and state cooperation that contributes to ensuring security and helping victims. They have shown that volunteering can be an effective tool in the fight against war and its consequences. Many volunteer organizations provided free medical care, took on the responsibility of supplying the military with necessary items and military equipment, organized the evacuation of people from the war zone, and provided psychological support to the military and their families. Conclusions.Volunteering in Ukraine is not a new phenomenon, but the war with Russia has become a catalyst for its development. This is a great example of how crises can encourage people to act together and interact to achieve a common goal. Volunteerism in Ukraine demonstrates the importance of spiritual unity and cooperation in difficult times, and is one of the most striking examples of civil society in action. The ideological principles of volunteering in Ukraine include non-profitability, openness, spirituality, and humanism, transparency and openness, security, and cooperation with government agencies and the military. The volunteer movement in Ukraine, especially in times of war, is an important factor in ensuring security and helping victims. Involvement of different groups of the population in volunteer activities, as well as cooperation with state and international structures, are important to ensure the effectiveness and success of volunteer activities. (shrink)

With the rapid development of society and the dramatic change of environment, previous career counseling focusing on personal choice has been difficult to meet individuals’ needs. It is very meaningful and valuable to introduce the ideology of Life Design Counseling. In this mini review, we introduce and analyze the theory and methodology of LDC. This review puts forward challenges in the field of LDC, including the lack of attention to clients from multiple backgrounds and professional counselors, the lack of diversified (...) methods in the intervention process, and the lack of diversified research. The theoretical research, practical research, and the integration of theory and practice of LDC still need to be further concerned by researchers. (shrink)

Ghazzālī; 1058-1111; Ibn Taymīyah, Aḥmad ibn ʻAbd al-Ḥalīm, 1263-1328; criticism and interpretation; Islam; doctrines.

On the concept of Mokṣa in contemporary Indic philosophy.

Autumn 1918 p. the breakup of the war-weakened Austro-Hungarian Empire was evident. Taking into account the existing claims of the Poles on the territory of Galicia, the first task of the Ukrainian parliamentary representation was to prepare for the re-election of the authorities in the region. On his initiative October 18-19, 1918 p. a constituent was held in Lviv - "a representative assembly of Ukrainian ambassadors to the parliament and regional councils of Galicia and Bukovina, bishops, delegates of Ukrainian parties...".

Andrey Sheptytsky - Metropolitan of the Greek Catholic Church, belongs to the key figures of Ukrainian history of the first half of the twentieth century. Its influence on the spiritual and national-political life of Ukrainians of that time, the processes of crystallization of national consciousness, as well as on the sphere of practical politics, cannot be overestimated. A. Sheptytsky's life path is an object of constant attention of historians, but it would still be early to assert its full and comprehensive (...) coverage. One of the debating issues in historiography remains the formation of national-political and religious views of the future Metropolitan. As you know, A. Sheptytsky was descended from the Polonized family, the process of his upbringing and education took place in the circle of Polish culture and national-state tradition. Already at adulthood, he made a conscious choice in favor of the Ukrainian Church and nation. The reasons and motives for this choice require a thorough study, taking into account the many factors that led it. (shrink)

A special place in A. Sheptytsky's socio-political activity is occupied by parliamentary activity. Considering the fact that the Metropolitan has been intolerant of the use of terror in political struggle, it becomes clear why parliamentary methods of struggle were most effective for him.

In this paper, an active frequency control strategy of wind turbines based on model predictive control is proposed by using the power margin of wind turbines operating in load shedding mode. The frequency response model of the microgrid system with the load shedding of the wind turbines is used to predict the output power and system frequency deviation of the wind turbine. According to the prediction information, the output power control signal of the model predictive controller in the wind turbine (...) can be optimized. On this basis, a wind turbine active participation frequency control strategy based on model predictive control is designed by rolling prediction and optimization. The wind turbine power control signal after the strategy is used to adjust the output power of the wind turbine and balance the change of the active power of the system to reduce the frequency deviation. (shrink)

Communitarianism has been misunderstood. According to some of its proponents, it supports the ‘Asian values’ argument that rights are incompatible with communitarian Asia because it prioritises the collective interest over individual rights and interests. Similarly, its critics are sceptical of its normative appeal because they believe that communitarianism upholds the community’s wants and values at all costs. I dispel this misconception by providing an account of communitarianism, properly understood. It is premised on the idea that we are partially constituted by (...) our communal attachments, or constitutive communities, which are a source of value to our lives. Given the partially constituted self, communitarianism advances the thin common good of inclusion. In this light, communitarianism, properly understood, is wholly compatible with rights, and is a potent source of solutions to controversial issues that plague liberal societies, such as the right of a religious minority to wear its religious garment in public. (shrink)

Classical supercommentary on Brahmasūtrabhāṣya of Madhva, 13th century.

Oholekha Yaʻaḳov -- Et amalenu -- Melekhet H.

Attending cram school has long been a trend in ethnic Chinese culture areas, including Taiwan. Despite the fact that school reform policies have been implemented in Taiwan, cram schools have continued to prosper. Therefore, in this educational culture, how to achieve a good educational effect is also a topic worthy of discussion. However, whether students really engage in those tutoring programs provided by cram schools has seldom been studied. To address this gap, this study explored how parents’ hovering attitude toward (...) life and coursework influences their children’s engagement in cram schools. This study targeted those students who attend English cram schools to test the correlates between two types of helicopter parenting, tutoring engagement and continued attendance at cram schools. A total of 320 questionnaires were sent out, and 300 were returned, giving an overall response rate of 93.75%. Excluding seven incomplete or invalid questionnaires, 293 valid questionnaires were received. The results of this study show that hovering behavior awareness is negatively related to cram school engagement, whereas cram school engagement is positively related to the intention to continue attending cram school. Moreover, the results imply that parents should alleviate their helicoptering behavior to enhance their children’s engagement in cram school tutoring programs. (shrink)

Study on Yoga philosophy of Patañjali in the light of Upanishads, Hindu philosophical classics.

Analytical study of the meaning of Intellect (Buddhi) with reference to Indic philosophy in relations to Sanskrit grammar.