Transcendental Number Theory Research Articles

Attractors are not algebraic

The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.

Penerapan Asuhan Keperawatan Berdasarkan Self Transcendence Theory Pamela G.Reed

Nursing care is a process or series of activities in nursing practice that is provided directly to clients/patients in various health service settings. In implementing nursing care based on self-transcendence theory, nurses play the role of facilitators to help individuals experience growth and development through the self-transcendence process. Self-transcendence theory was developed by Pamela G. Reed and emphasizes the spiritual and transpersonal aspects of treatment. Reed explained that nurses act as facilitators to explore positive things and build bio-psycho-social spiritual well-being. Self Transcendence Theory aims to understand a person's movement in transcending self-limitations and achieving higher well-being. This theory focuses on five concepts, namely vulnerability, self-transcendence, well-being, moderating-mediating factors, and point of intervention. This self-transcendence offers the best for living in peace and prosperity. Apart from that, this theory is also used in the context of counseling services to help individuals utilize inner resources that empower patient capacities. Thus, the aim of the Self Transcendence theory is to understand and encourage individuals to achieve higher well-being through holistic self-understanding and development.

The development of P-adic numbers theoretically and their use in number theory

Stemming from the need to formalize the divisibility of rational numbers, p-adics provide a unique mathematical perspective rooted in prime number theory. This article provides a comprehensive discussion of p-adic numbers, including their research background, definition, properties, theory, and extensions. This study first elucidates the historical background and significance of p-adic numbers, emphasizing their key role in number theory and its applications. At the heart of this research is a deep dive into the precise definition of p-radical numbers, revealing their unique, often counterintuitive, distance measure. We look at their fundamental characteristics, demonstrate their illogical characteristics, and discuss how they affect transcendental number theory, Diophantine equations, and algebraic number theory. Furthermore, this article explores the theoretical foundations of p-radical numbers and their extensions, emphasizing their integral role in advanced mathematical structures such as p-radical analysis and p-radical geometry. By covering these aspects, this study aims to highlight the lasting impact of p-adic on modern mathematics, reshape our understanding of divisibility, and advance mathematical inquiry into new and uncharted territory.

Development and Validation of the Transcendent Beliefs Scale (TBS): A 12-Item Assessment of Self-Transcendent States

Objective: The purpose of this study was to develop and validate the Transcendent Beliefs Scale (TBS). We defined transcendent experiences as an experience of unity with others and the whole of existence, which is associated with behavioral, cognitive, emotional, and transcendental factors. Our aim was to develop a very brief instrument that accurately reflected the phenomenology of transcendent states. Keywords: trauma, chronic pain, dissociation, EFT, EMDR, mindful breathing, bilateral stimulation, BLS, WHEE, HeartMath, blended, combined, interventions Method: The factor structure, validity, and reliability of the TBS were assessed across a sample of university students and energy therapy practitioners (N = 305). Results: With respect to convergent validity, we found that transcendent experiences were positively correlated with daily spiritual experiences, trait hope, and meaning in life. Discriminant validity was demonstrated through a negative weak relationship to narcissistic personality traits. Exploratory factor analysis reduced the scale to 12 items and a two-factor model was found to fit the data. Conclusion: The results indicate that the TBS is a valid and reliable scale to measure transcendent beliefs in the general population. It is suitable for research projects in which a brief measure of transcendence is required. Keywords: transcendent beliefs, transcendent experiences, universal oneness, scale validation, unity consciousness, existence, transcendental phenomenology.

Exponential sums equations and tropical geometry

Zilber’s Exponential-Algebraic Closedness Conjecture states that algebraic varieties in {{mathbb {C}}}^n times ({{mathbb {C}}}^times )^n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of {textrm{exp}}. We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry.

Interpretation of the Concept of Social Constructivism by L. S. Vygotsky through the Semiotics of Education, the Theory of Transcendence in Education and the Theology of Personality by V. N. Lossky

The subject of the work is the theses of the concept of developmental learning, semiotics of education, theory of transcendence in education, and theology of personality. The aim of the research is to develop a practical pedagogical method based on the discussed concepts. The author demonstrates the key positions of the concept of developmental learning, such as the zone of proximal development and scaffolding; reveals the understanding of education as a semiotic system and the participation of transcendent reality in it; and outlines the main theses of the theology of personality by V.N. Lossky. Then the author combines the aforementioned theories to create a pedagogical method, which he defines as an integration of the developmental learning approach with the position of the theology of personality. The author's special contribution is the generalization of scattered concepts into unified conclusions and the development of a practical pedagogical method, which includes a number of theses, specific components of a case for conducting a lesson as a template, and instructions for implementing the case in accordance with the generally accepted stages of a lesson also in the form of a template. The author concludes that the developed case is universal and can be applied at different levels of education. The author also notes the value of the material based on which the method was developed for further study of each of the presented areas of science. The author emphasizes that the application of the method is not limited to the religious context, as the method was developed based on non-religious research, and the positions of Orthodox theology of personality correspond to the presented theories of modern philosophy of education.

Higher Ramanujan Equations and Periods of Abelian Varieties

We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series E 2 E_2 , E 4 E_4 , E 6 E_6 . Such “higher Ramanujan equations” are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford’s theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing ( E 2 , E 4 , E 6 ) (E_2,E_4,E_6) , which are also shown to be defined over Z \mathbf {Z} . This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko’s celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck’s Period Conjecture. Working in the complex analytic category, we prove “functional” transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations.

Реконструкция трех типов трансценденций в социологии религии Томаса Лукмана

Although Thomas Luckmann was at the origin of secularization theory, which is considered to have finally lost its relevance in the early 1990s, some parts of his theory of religion are still of interest. This paper focuses on the transcendence concept, which is assumed to be crucial for Luckmann’s theory of religion and remains a relevant resource for the study of religion today. The text advances and substantiates the thesis that explicitly or implicitly, three types of transcendence can be found in Lukman’s theory of religion. The description of the anthropological transcendence focuses on the transcending of the human organism to the social world. The explication of this process leads us to the conclusion that the condition for the possibility of religion is the property of the organism to construct gradients of importance before any reflection on this importance. The description of phenomenological transcendence focuses on the process of “shrinking” of the experience of the transcendent underlying religion: from religions of “great” transcendence (world religions), to religions of “medium” or even “little” transcendence (privatized religions). The discussion of this type leads to the idea that religious experience does not disturb the course of everyday life, but is a condition of it, and that although any experience can be the basis of religion, there must be a way of selecting experiences that have a higher probability of being experienced religiously. The description of sociological transcendence focuses on the process of internalization and reconstruction of self-evident “ultimate” socially objectified significances (socio-historical a priori) transcendent to the individual and underlying their everyday life. In this sense, the process of privatization of religion suggests that the practice of “choosing” a private religion is part of, rather than a condition of, a system of “ultimate” significance. Understood in this way, Lukman’s theory of transcendence allows us to look at some relationships important for the sociology of religion from a different angle. For example, the relationship between religion in the public and private sphere, the relationship between religious and everyday experience, and the relationship between the functional boundaries of religious and secular institutions.

Polynomial dynamics and local analysis of small and grand orbits

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.

Kant’s Theory of Human Transcendence (Ontology) and its Pitfalls

Purpose: This work offers a caveat regarding the human propensity to error, even in situations of very meticulous and seemingly thorough philosophical investigations.
 Methodology: This paper is a critical, analytic and evaluative, in its exposition of Kant’s ontology.
 Findings: Kant already had a theory of the origin of concepts which reinforced his theory of the nature of judgments. Based on his acceptance of the Cartesian psychology of perception, he claimed that man thinks only thoughts and perceives only perceptions; and that the mind is aware only of itself and its own states. The situation puts in relief what finally is an issue between a theory which is pinned to a belief in an absolute beyond history and behind experience, and one which is frankly experimental.
 Unique Contribution to theory, practice and policy (recommendation): In a manner that is both instructive and corrective, this work contributes to the extant repertoire of discussions on Kant’s ontology. It offers a significantly demanding ratiocinative interrogation of the foundations upon which Kant’s theory of human transcendence was premised or predicated.

The sad clown paradox: A theory of comic transcendence

One prevailing cultural sensibility of our time is a concern with the state of our mental health, another is our obligation and desire for a good ‘sense of humour’. At present the two are conflated, most often through a long-standing cultural trope: the sad clown paradox – those who make us laugh the most tend to be the most prone to mental health problems. This article views the ‘sad clown paradox’ as less about the peculiarity or exceptional status of a comedian's mind, more about how the cognitive burdens of modernity are rendered bearable and collectively recognised in thought and sentiment by humour. Accounts of comedy and mental health conflate good comedy with mental anguish. By unpacking this knotted relationship, it is argued that comedian's humour performs a way for contemporary people to deal with modernity's fragmentary character of life from within their own inner worlds and selves.

Chinese Traditional Bioethics.

Chinese traditional bioethics is based on the life origin theory of "Qi (vital force) permeates everything under heaven", the attitude towards life theory of "loving people" and "loving things", and the life transcendence theory of "eternity" and "immortality". The life origin theory answers the question of how life comes into being, which is the basis for exploring the mystery of life. The attitude towards life theory gives answers to the question of how to treat life properly and establishes the basic requirements and criteria for treating life. The life transcendence theory responds to the question of how to make life more meaningful. Those three thoughts support each other and explain each other with the inner logical band between these three basic thoughts, and tentatively offers a reasonable way for the reconstruction of traditional Chinese bioethics, which can be a special theoretical resource for today's bioethical research.

O-Minimal Invariants for Discrete-Time Dynamical Systems

Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants , which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory.

Counting rational points on elliptic curves with a rational 2-torsion point

Let E/\mathbb{Q} be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if E has full rational 2-torsion, the number N_E(B) of rational points with Weil height bounded by B is \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) . In this paper we exploit the method of descent via 2-isogeny to extend this result to elliptic curves with just one nontrivial rational 2-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound N_E(B) = \operatorname{exp}\big(O \big(\frac{\operatorname{log}B}{\sqrt{\operatorname{log}\operatorname{log} B}}\big)\big) for these curves and to remove a deep transcendence theory ingredient from the proof.

Algebraic and Model Theoretic Properties of O-minimal Exponential Fields

AbstractAn exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1,<)$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group of positive elements $(K^{>0},\cdot ,1,<)$ . The structure $(K,+,-,\cdot ,0,1,<,\exp )$ is then called an ordered exponential field (cf. [6]). A linearly ordered structure $(M,<,\ldots )$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0) = 1$ . This study is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0)=1$ is elementarily equivalent to $\mathbb {R}_{\exp }$ .Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $(\mathbb {R},+,-,\cdot ,0,1,<,\exp )$ , where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$ . Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot ,0,1, <,\exp \}$ holds for $(K,+,-,\cdot ,0,1,<,\exp )$ if and only if it holds for $\mathbb {R}_{\exp }$ .The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$ . To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$ . However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$ (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$ .Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: sebastian.krapp@uni-konstanz.deURL: https://d-nb.info/1202012558/34

The colonization of epidemiological knowledge: a decolonial reading of the contemporaneity of the COVID-19 pandemic

This paper makes a critical assessment of epidemiology with the COVID-19 pandemic as a social event. It examines the philosophical reflection in which Agamben defines as contemporary those able to stand back to see the dark side of their own era. In the light of decolonial criticism, the concept of "epidemiological transition," with its theory of transcendence of "social determinants of health" and binarism of epidemiological variables as supports of the biomedical and quantitative structuring of the epidemiology of risk factors is queried. The scientific ambition to dominate nature and the engendering of a linear and evolutionary historical time, beginning in western modernity, contextualizes the epistemicides of popular wisdom and the coloniality of epidemiological knowledge. The theoretical constitution of decolonial thought is historically analyzed, highlighting its greater critical potential to reveal the structural colonization of epidemiological knowledge. The post-pandemic future is considered and Prigogine's idea of bifurcation - as elaborated by Sousa Santos - and Paulo Freire's untested feasibility are related with the concept of time as the creation and expectation of social transformation.

Ax–Schanuel type theorems on functional transcendence via Nevanlinna theory

In this paper, we apply Nevanlinna theory to prove two Ax–Schanuel type theorems for functional transcendence when the original exponential map is replaced by other meromorphic functions. We give examples to show that these results are optimal. As a byproduct, we also show that analytic dependence implies algebraic dependence for certain classes of entire functions. Finally, some links to transcendental number theory and geometric Ax–Schanuel theorem will be discussed.

Undecidable arithmetic properties of solutions of Fredholm integral equations

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f(α) is transcendental or algebraic if f(z) is an E-function and α∈Q‾⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f(z) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f(0)∈Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6].

Non-commutative analytic subgroup theorem

In this note we derive a non-commutative version of the Wüstholz analytic subgroup theorem in transcendence theory from the original (commutative) theorem and provide an application.

Pluralists and Pantheists: Spinoza, Deleuze and the non-fiction of D.H. Lawrence

D.H. Lawrence was not just a novelist, poet or travel writer. He was deeply interested in philosophy, and even went so far as to write a serious book of philosophy called At the Gates, which was unfortunately lost. What follows is an attempt to explain what D.H. Lawrence meant when he said: ‘The Greeks, being sane, were pluralists and pantheists. And so am I’. The way I propose to do this is by looking at the striking similarities between Lawrence, Spinoza and Deleuze. I wish to show that there is a kind of dynamic in Spinoza’s theory of knowledge, which can clarify the relationship between pluralism and pantheism. What is particularly interesting in this respect is that Deleuze comments on Lawrence in some detail in his Spinoza lectures at Vincennes. Deleuze explicitly uses Lawrence’s non-fictional writings to illustrate and elucidate Spinoza’s three types of knowledge. Ultimately, Lawrence’s pluralism leads him to a radical criticism of the inadequacy of his contemporaries’ thought, as well as an emphasis on the relationships and correspondences that constitute our daily experience. It also goes a long way to explaining his pantheism, and beyond this, to his theory of transcendence and the eternal.