In this letter we demonstrate that the fermionic zero modes on a superconducting domain wall can be associated to an one dimensional \ supersymmetry that contains non-trivial topological charges. In addition, the system also possesses three distinct \ supersymmetries with non-trivial topological charges and we also study some duality transformations of the supersymmetric algebras.

The organization of the genome into topologically associated domains (TADs) appears to be a fundamental process occurring across a wide range of eukaryote organisms, and it likely plays an important role in providing an architectural foundation for gene regulation. Initial studies emphasized the remarkable parallels between TAD organization in organisms as diverse as Drosophila and mammals. However, whereas CCCTC‐binding factor (CTCF)/cohesin loop extrusion is emerging as a key mechanism for the formation of mammalian topological domains, the genome organization (...) in Drosophila appears to depend primarily on the partitioning of chromatin state domains. Recent work suggesting a fundamental conserved role of chromatin state in building domain architecture is discussed and insights into genome organization from recent studies in Drosophila are considered. (shrink)

Topologically associating domain (TAD) boundaries are the flanking edges of TADs, also known as insulated neighborhoods, within the 3D structure of genomes. A prominent feature of TAD boundaries in mammalian genomes is the enrichment of clustered CTCF sites often with mixed orientations, which can either block or facilitate enhancer–promoter (E‐P) interactions within or across distinct TADs, respectively. We will discuss recent progress in the understanding of fundamental organizing principles of the clustered CTCF insulator codes at TAD boundaries. Specifically, (...) both inward‐ and outward‐oriented CTCF sites function as topological chromatin insulators by asymmetrically blocking improper TAD‐boundary‐crossing cohesin loop extrusion. In addition, boundary stacking and enhancer clustering facilitate long‐distance E‐P interactions across multiple TADs. Finally, we provide a unified mechanism for RNA‐mediated TAD boundary function via R‐loop formation for both insulation and facilitation. This mechanism of TAD boundary formation and insulation has interesting implications not only on how the 3D genome folds in the Euclidean nuclear space but also on how the specificity of E‐P interactions is developmentally regulated. (shrink)

Recent investigations have revealed 1) that the isochores of the human genome group into two super‐families characterized by two different long‐range 3D structures, and 2) that these structures, essentially based on the distribution and topology of short sequences, mold primary chromatin domains (and define nucleosome binding). More specifically, GC‐poor, gene‐poor isochores are low‐heterogeneity sequences with oligo‐A spikes that mold the lamina‐associated domains (LADs), whereas GC‐rich, gene‐rich isochores are characterized by single or multiple GC peaks that mold the topologically (...) associating domains (TADs). The formation of these “primary TADs” may be followed by extrusion under the action of cohesin and CTCF. Finally, the genomic code, which is responsible for the pervasive encoding and molding of primary chromatin domains (LADs and primary TADs, namely the “gene spaces”/“spatial compartments”) resolves the longstanding problems of “non‐coding DNA,” “junk DNA,” and “selfish DNA” leading to a new vision of the genome as shaped by DNA sequences. (shrink)

We discuss here a series of testable hypotheses concerning the role of chromosome folding into topologically associating domains (TADs). Several lines of evidence suggest that segmental packaging of chromosomal neighborhoods may underlie features of chromatin that span large domains, such as heterochromatin blocks, association with the nuclear lamina and replication timing. By defining which DNA elements preferentially contact each other, the segmentation of chromosomes into TADs may also underlie many properties of long‐range transcriptional regulation. Several observations suggest that TADs (...) can indeed provide a structural basis to regulatory landscapes, by controlling enhancer sharing and allocation. We also discuss how TADs may shape the evolution of chromosomes, by causing maintenance of synteny over large chromosomal segments. Finally we suggest a series of experiments to challenge these ideas and provide concrete examples illustrating how they could be practically applied. (shrink)

TAD boundaries are genomic elements that separate biological processes in neighboring domains by blocking DNA loops that are formed through Cohesin‐mediated loop extrusion. Most TAD boundaries consist of arrays of binding sites for the CTCF protein, whose interaction with the Cohesin complex blocks loop extrusion. TAD boundaries are not fully impermeable though and allow a limited amount of inter‐TAD loop formation. Based on the reanalysis of Nano‐C data, a multicontact Chromosome Conformation Capture assay, we propose a model whereby clustered CTCF (...) binding sites promote the successive stalling of Cohesin and subsequent dissociation from the chromatin. A fraction of Cohesin nonetheless achieves boundary read‐through. Due to a constant rate of Cohesin dissociation elsewhere in the genome, the maximum length of inter‐TAD loops is restricted though. We speculate that the DNA‐encoded organization of stalling sites regulates TAD boundary permeability and discuss implications for enhancer–promoter loop formation and other genomic processes. (shrink)

Deletions, duplications, insertions, inversions, and translocations, collectively called structural variations (SVs), affect more base pairs of the genome than any other sequence variant. The recent technological advancements in genome sequencing have enabled the discovery of tens of thousands of SVs per human genome. These SVs primarily affect non‐coding DNA sequences, but the difficulties in interpreting their impact limit our understanding of human disease etiology. The functional annotation of non‐coding DNA sequences and methodologies to characterize their three‐dimensional (3D) organization in the (...) nucleus have greatly expanded our understanding of the basic mechanisms underlying gene regulation, thereby improving the interpretation of SVs for their pathogenic impact. Here, we discuss the various mechanisms by which SVs can result in altered gene regulation and how these mechanisms can result in rare genetic disorders. Beyond changing gene expression, SVs can produce novel gene‐intergenic fusion transcripts at the SV breakpoints. (shrink)

Recent advances in genomic and imaging techniques have revealed the complex manner of organizing billions of base pairs of DNA necessary for maintaining their functionality and ensuring the proper expression of genetic information. The SMC proteins and cohesin complex primarily contribute to forming higher‐order chromatin structures, such as chromosomal territories, compartments, topologically associating domains (TADs) and chromatin loops anchored by CCCTC‐binding factor (CTCF) protein or other genome organizers. Cohesin plays a fundamental role in chromatin organization, gene expression and regulation. (...) This review aims to describe the current understanding of the dynamic nature of the cohesin‐DNA complex and its dependence on cohesin for genome maintenance. We discuss the current 3C technique and numerous bioinformatics pipelines used to comprehend structural genomics and epigenetics focusing on the analysis of Cohesin‐centred interactions. We also incorporate our present comprehension of Loop Extrusion (LE) and insights from stochastic modelling. (shrink)

Information management systems improve the retention of information in large collections. As such they act as memory prostheses, implying an ideal basis in human memory models. Since humans process information by association, and situate it in the context of space and time, systems should maximize their effectiveness by mimicking these functions. Since human attentional capacity is limited, systems should scaffold cognitive efforts in a comprehensible manner. We propose the Principles of Mnemonic Associative Knowledge (P-MAK), which describes a framework for semantically (...) identifying, organizing, and retrieving information, and for encoding episodic events by time and stimuli. Inspired by prominent human memory models, we propose associative networks as a preferred representation. Networks are ideal for their parsimony, flexibility, and ease of inspection. Networks also possess topological properties—such as clusters, hubs, and the small world—that aid analysis and navigation in an information space. Our cognitive perspective addresses fundamental problems faced by information management systems, in particular the retrieval of related items and the representation of context. We present evidence from neuroscience and memory research in support of this approach, and discuss the implications of systems design within the constraints of P-MAK’s principles, using text documents as an illustrative semantic domain. (shrink)

Recent years have witnessed an explosion of the single-cell biochemical toolbox including chromosome conformation capture -based methods that provide novel insights into chromatin spatial organization in individual cells. The observations made with these techniques revealed that topologically associating domains emerge from cell population averages and do not exist as static structures in individual cells. Stochastic nature of the genome folding is likely to be biologically relevant and may reflect the ability of chromatin fibers to adopt a number of alternative (...) configurations, some of which could be transiently stabilized and serve regulatory purposes. Single-cell Hi-C approaches provide an opportunity to analyze chromatin folding in rare cell types such as stem cells, tumor progenitors, oocytes, and totipotent cells, contributing to a deeper understanding of basic mechanisms in development and disease. Here, we review key findings of single-cell Hi-C and discuss possible biological reasons and consequences of the inferred dynamic chromatin spatial organization. Single-cell Hi-C expands the molecular biology toolbox for studies of chromatin structure in individual cells. This technique allows one to analyze cell-to-cell variability in TAD and loop structure, and to capture chromosome conformation in rare cell types such as oocytes, totipotent cells, and tumor progenitors. (shrink)

Insulators play a central role in subdividing the chromosome into a series of discrete topologically independent domains and in ensuring that enhancers and silencers contact their appropriate target genes. In this review we first discuss the general characteristics of insulator elements and their associated protein factors. A growing collection of insulator proteins have been identified including a family of proteins whose expression is developmentally regulated. We next consider several unexpected discoveries that require us to completely rethink how insulators (...) function (and how they can best be assayed). These discoveries also require a reevaluation of how insulators might restrict or orchestrate (by preventing or promoting) interactions between regulatory elements and their target genes. We conclude by connecting these new insights into the mechanisms of insulator action to dynamic changes in the three‐dimensional topology of the chromatin fiber and the generation of specific patterns of gene activity during development and differentiation. (shrink)

The concept of representation has been a key element in the scientific study of mental processes, ever since such studies commenced. However, usage of the term has been all but too liberal—if one were to adhere to common use it remains unclear if there are examples of physical systems which cannot be construed in terms of representation. The problem is considered afresh, taking as the starting point the notion of activity spaces—spaces of spatiotemporal events produced by dynamical systems. It is (...) argued that representation can be analyzed in terms of the geometrical and topological properties of such spaces. Several attributes and processes associated with conceptual domains, such as logical structure, generalization and learning are considered, and given analogues in structural facets of activity spaces, as are misrepresentation and states of arousal. Based on this analysis, representational systems are defined, as is a key concept associated with such systems, the notion of representational capacity. According to the proposed theory, rather than being an all or none phenomenon, representation is in fact a matter of degree—that is can be associated with measurable quantities, as is behooving of a putative naturalistic construct. (shrink)

In this dissertation we explore projective Fraïssé theory and its applications, as well as limitations, to the study of compact metrizable spaces. The goal of projective Fraïssé theory is to approximate spaces via classes of finite structures and glean topological or dynamical properties of a space by relating them to combinatorial features of the associated class of structures. Using the framework of compact metrixable structures, we establish general results which expand and help contextualize previous works in the field. Many (...) proofs in the domain of projective Fraïssé theory are carried out in a context dependent fashion and have thus far eluded clean generalizations. A reason is to be found in the lack of a clear understanding of which spaces are amenable to be studied via projective Fraïssé limits. We give both positive and negative results in this direction. We isolate a combinatorial condition which entails a correspondence between finite structures and regular quasi-partitions of compact metrizable spaces. This correspondence greatly aids the combinatorial-toplogical translation. We apply this machinery to study a class of one-dimensional compact metrizable spaces, which we call smooth fences. To this end, we isolate a class of finite structures—finite partial orders whose Hasse diagram is a forest—whose projective Fraïssé limit approximates a distinctive smooth fence with remarkable properties. We call it the Fraïssé fence and characterize it topologically by carefully exploiting the bridge between the combinatorial and topological worlds. We explore homogeneity and universality features of the Fraïssé fence and the properties of its spaces and endpoints, and provide some results on the dynamics of its group of homeomorphisms. The dissertation is partially based on [1, 2]. (shrink)

We explore the following two cognitive questions regarding crosslinguistic variation in lexical semantic systems: Why are some linguistic categories—that is, the associations between a term and a portion of the semantic space—harder to learn than others? How does learning a language‐specific set of lexical categories affect processing in that semantic domain? Using a computational word‐learner, and the domain of color as a testbed, we investigate these questions by modeling both child acquisition of color terms and adult behavior on (...) a non‐verbal color discrimination task. A further goal is to test an approach to lexical semantic representation based on the principle that the more languages label any two situations with the same word, the more conceptually similar those two situations are. We compare such a crosslinguistically based semantic space to one based on perceptual similarity. Our computational model suggests a mechanistic explanation for the interplay between term frequency and the semantic closeness of learned categories in developmental error patterns for color terms. Our model also indicates how linguistic relativity effects could arise from an acquisition mechanism that yields language‐specific topologies for the same semantic domain. Moreover, we find that the crosslinguistically inspired semantic space supports these results at least as well as—and in some aspects better than—the purely perceptual one, thus confirming our approach as a practical and principled method for lexical semantic representation in cognitive modeling. (shrink)

Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, (...) I address two criticisms that have been raised in Tatton-Brown against our approach: that it leads to a form of relativism according to which validity is equated with social agreement and that it implies an antiformalizability thesis according to which it is not the case that all rigorous mathematical proofs can be formalized. I reject both criticisms and suggest that our previous case studies provide insight into the plausibility of two related but quite different theses. (shrink)

This paper is concerned with certain ontological issues in the foundations of geographic representation. It sets out what these basic issues are, describes the tools needed to deal with them, and draws some implications for a general theory of spatial representation. Our approach has ramifications in the domains of mereology, topology, and the theory of location, and the question of the interaction of these three domains within a unified spatial representation theory is addressed. In the final part we also consider (...) the idea of non-standard geographies, which may be associated with geography under a classical conception in the same sense in which non-standard logics are associated with classical logic. (shrink)

In this thesis, we study the complexity of some mathematical problems: in particular, those arising in computable analysis and algorithmic learning theory for algebraic structures. Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example descriptive set theory and reverse mathematics.After giving the necessary preliminaries, we first study the uniform computational strength of the Cantor–Bendixson theorem in (...) the Weihrauch lattice. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of $\mathbf {\Pi }^1_1\mathsf {-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.We continue considering problems related to (induced) subgraphs. We provide results on the (effective) Wadge complexity of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph G, and they take as input a graph H, answering whether G is an (induced) subgraph of H: we also consider the opposite problem (i.e., answering whether H is an induced subgraph of G). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph G, and they take as input a graph H such that G is an (induced) subgraph H: the output is a copy of G in H. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.We then move our attention to algorithmic learning theory, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures $\mathfrak {K}$, a learner receives larger and larger pieces of an arbitrary copy of a structure in $\mathfrak {K}$ and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that $\mathfrak {K}$ is learnable if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on infinite binary sequences and then, replacing $E_0$ with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of E-learnability, where a family of structures $\mathfrak {K}$ is E-learnable, for a Borel equivalence relation E, if there is a continuous reduction from the isomorphism relation associated with $\mathfrak {K}$ to E. It is then natural to ask how the notion of E-learnability interacts with “classical” learning paradigms.Finally, we study the number of mind changes that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of Turing reducibility) affects the number of mind changes for learning a given family.Abstract prepared by Vittorio Cipriani.E-mail: vittorio.cipriani17@gmail.com.URL: https://air.uniud.it/handle/11390/1252226. (shrink)

It has been argued within some philosophy of quantum gravity circles that endorsing Lewisian modal metaphysics is incompatible with endorsing the fundamental physical ontology of any quantum gravity theory. Speaking concisely, the unsolvable tension would be between Lewis' metaphysical commitment to the fundamentality of space and time, and the physical lesson of quantum gravity about the disappearance of space and time from the fundamental structure of the world. In this essay I argue against the idea that the tension is unsolvable. (...) The analysis does not apply to quantum gravity in general, but only to the specific perspective delivered by quantum string theory. In the first two sections I describe what I think is the general formal payoff of Lewisian possible worlds semantics, and more importantly the general metaphysical payoff generated by some Lewisian metaphysical applications of that formal framework. Lewisian possible worlds semantics is not among the topics of this essay, still a concise presentation of its basic features is required for a more accurate description of the type of modal metaphysics Lewis delivers by applying that formal machinery. The moral of the story in the first two sections is that an attempt of preserving crucial features of Lewisian modal metaphysics, although endorsing the quantum string theory lesson on spacetime non fundamentality, would be not only possible, but also philosophically worthy in a general sense. Then, section three is divided in three parts. In the first part I summarize some formal tools of the methodology used elsewhere - (Vistarini, Routledge, 2019, chapter 6)- to argue for string theory background independence. More precisely, I briefly re-describe some crucial features of a topological-vector bundle structure there defined on a "space" associated to the theory. Such structure is there shown to carry physical and dynamical information about quantum strings systems supporting the thesis of background independence of quantum strings laws (string theory background independence is not a topic of this essay, and the construction of the full formal structure as well as the full argumentative line can be found in the work cited above). The rationale behind the choice of including these formal tools is explained in the second and third parts of section three: that very same topological-fiber bundle structure associated to string theory, if read differently, is shown to also carry structural properties of the Lewisian metaphysical scheme. More precisely, in the second part of section three I attempt to show that such topological-fiber bundle structure owns a topology which qualifies as a plausible accessibility structure extending the Lewisian similarity accessibility. To avoid confusion, I will denote the extended similarity order with S*, or S*-similarity. The latter has some interesting properties. It is quantitatively describable, and even more interestingly its spectrum of variation is a countable one. Then, insofar Lewisian similarity order can be regained from the S*-similarity, one might say that the latter is a metaphysical accessibility relation underlying the former because "more fundamental". As we will see, the explanation of what "more fundamental" means in this metaphysical context mainly involves the idea that the formal nature of S*-similarity, in particular its property of having a countable spectrum of variation, calibrates a metaphysical similarity that reflects the discreteness of the fundamental quantum nature of reality. The extended modal metaphysics with accessibility structure S*-similarity is informed by quantum string physics. Within the extended scheme, the Lewisian claim "things could have been different in countless ways" (Lewis 1973) is an approximate one because only permitted by ordinary language. As soon as one refines the accessibility structure by using the formal language of quantum string theory, it appears things could have been different in countable ways. But the Lewisian variety of alternative states of affairs can be regained from the underlying countable variety via a specific interpretation of the Humean supervenience thesis, one that changes the metaphysical status of the thesis' content and restricts the thesis' domain of application. Finally, in the third part of section three I attempt to show that the same topological-fiber bundle structure used to extend Lewisian similarity has a bundle component producing the formal scheme for a tentative extension of Lewisian nomological accessibility through the lens of string dualities. This will prove to be a non trivial task, and the Lewisian accessibility structure will be regained from the extended one only partially. Finally, the essay contains two short appendices summarizing two argumentative lines outside the main topic of the essay. They briefly describe different ways in which the non fundamentality of spatiotemporal structures can be found in quantum string theory. These sections can be skipped. Nevertheless, they might contribute to understand more robustly the general view endorsed by this essay, i.e. that there isn't any unsolvable tension between Lewis' commitment to the fundamentality of spatiotemporal relations and quantum gravity's lesson about their non fundamentality. -/- . (shrink)

The method of Quasi-Analysis used by Carnap in his program of theconstitution of concepts from finite observations has the following twogoals: (1) Given unsharp observations in terms of similarity relations thetrue properties of the observed objects shall be obtained by a suitablelogical construction. (2) From a single relation on a finite domain,different dimensions of qualities shall be reconstructed and identified. Inthis article I show that with a slight modification Quasi-Analysis iscapable of fulfilling the first goal for a single observable (...) dimension. Weobtain a partition of the so-called Quality Classes representing thepairwise disjoint and exhaustive extensions associated to the ``values'''' ofthe observable. On the other hand, an example demonstrates that the methodfails, as Goodman has pointed out, for a relation expressing similarity withregard to at least one out of many properties.Since it seems to be impossible in general to reconstruct more-dimensionalqualities from a single similarity relation, the constitution of at least asmany similarity relations as there are qualities have to be presumed. Thenit is possible to state adequate sufficient conditions for the dimension ofthe observable space, even if some of the similarity relations might dependon others. The concept of topological dimension cannot be used for thispurpose on finite sets of observations. We replace it by a set-algebraicalcondition on the Quality Classes. (shrink)

Gene transcription in eukaryotes is associated with conformational changes of a large area of chromatin adjacent to a gene. This rearrangement may involve the whole loop (topological domain) to which a given gene belongs.Regulatory events associated with activation or inactivation of transcription are found to act through relatively short nucleotide sequences, often located several thousand base pairs apart from gene. These sequences, termed enhancers may act independently on their distance from or orientation with respect to the gene.Both (...) long‐range conformational rearrangements of chromatin and any long distance effects of enhancers may be associated with elastic torsional strain in DNA, arising in transcriptionally active chromatin loops. Enhancer effects, in particular, may be associated with conformational transitions in DNA competing for the energy deposited in the elastic spring of the DNA. (shrink)

Data Transfer Service (DTS) is an open-source project that is developing a document-centric message model for describing a bulk data transfer activity, with an accompanying set of loosely coupled and platform-independent components for brokering the transfer of data between a wide range of (potentially incompatible) storage resources as scheduled, fault-tolerant batch jobs. The architecture scales from small embedded deployments on a single computer to large distributed deployments through an expandable ‘worker-node pool’ controlled through message-orientated middleware. Data access and transfer efficiency (...) are maximized through the strategic placement of worker nodes at or between particular data sources/sinks. The design is inherently asynchronous, and, when third-party transfer is not available, it side-steps the bandwidth, concurrency and scalability limitations associated with buffering bytes directly through intermediary client applications. It aims to address geographical–topological deployment concerns by allowing service hosting to be either centralized (as part of a shared service) or confined to a single institution or domain. Established design patterns and open-source components are coupled with a proposal for a document-centric and open-standards-based messaging protocol. As part of the development of the message protocol, a bulk data copy activity document is proposed for the first time. (shrink)

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic (...) geometry a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

The great mathematician, physicist, and philosopher, Hermann Weyl, once called mathematics the “science of the infinite.” This is a fitting title: contemporary mathematics—especially Cantorian set theory—provides us with marvelous ways of taming and clarifying the infinite. Nonetheless, I believe that the epistemic significance of mathematical infinity remains poorly understood. This dissertation investigates the role of the infinite in three diverse areas of study: number theory, cosmology, and probability theory. A discovery that emerges from my work is that the epistemic role (...) of the infinite varies, often in surprising ways, across different domains of knowledge. -/- My first chapter examines the role of mathematical infinity in number theory. It is reasonable to think that theorems concerning finite patterns and structures in the natural numbers are particularly “simple” or “elementary.” Indeed, such statements are comprehensible to a wide range of investigators, regardless of their mathematical training. One might then expect proofs of these theorems to be similarly comprehensible. However, many proofs, especially those that utilize only finitary methods, are exceedingly difficult to understand. Consequently, one finds that finitary theorems are often re-proved using infinitary techniques. My claim is that this is because infinitary proofs are often explanatory, while finitary proofs are not. This chapter analyzes why this is the case. Along the way, I investigate other questions of long-standing interest in the philosophy of mathematics, e.g., the role of purity/impurity ascriptions and the nature of the content of a theorem. In particular, I diagnose the explanatory potential of the infinite by articulating a new construal of content. This new construal both saves intuitive epistemic ascriptions made in mathematical practice and explains the unexpected role of the infinite in providing explanatory proofs of finitary statements. Thus, in number theory, my claim is that the infinite often plays an explanatory role. -/- My second chapter turns to the role of the infinite in cosmology. It investigates a question much discussed by philosophers and physicists alike: is the spatial extent of the universe finite or infinite? Contemporary cosmological research has indicated that one of the essential determinants of the extent of the universe is the topology we ascribe to space. Topology is a global property, which may suggest that it is not testable through local observation. Nonetheless, some cosmologists have indicated that it may be empirically detected, thereby providing an answer to the question of spatial extent. I argue that, in fact, the epistemic status of the topology of space is extremely subtle and not well captured by any of the categories commonly employed by philosophers of science. In particular, I argue that topological properties are neither empirical nor a priori (even in suitably weakened senses). Furthermore, I claim that we should prefer topological properties that generate finite universe models (consistent with our best data) in order to avoid extremely thorny issues concerning the physics of an infinite universe. I argue for such a preference on the grounds of the simplicity and explanatory power of finite universe models. Thus, in cosmology, my claim is that the finite often plays an explanatory and simplifying role. -/- My third chapter investigates several paradoxes that arise in the foundations of infinitary probability theory: the Label Invariance Paradox, God’s Lottery, and Bertrand’s Paradox. I argue that these have been poorly understood because they do not expressly concern probability theory, but rather our intuitions about—and formal techniques for dealing with— infinite sets. The paradoxes in question are, in fact, symptoms of our complete reliance upon Cantorian cardinality and its associated criterion of sameness of “size.” That is, two sets have the same cardinality if and only if the elements of the sets can be placed in 1-1 correspondence. When applied to infinite sets, this criterion produces counterintuitive verdicts. For instance, given a fair lottery on the natural numbers, we expect that the probability of drawing an even number is 1/2, and likewise for drawing an odd number. However, one can construct “relabellings” of the naturals such that the probability of drawing an even number remains 1/2, while the probability for drawing an odd number becomes 1/4. I argue that, ultimately, it is the coarseness of Cantorian cardinality that generates the probabilistic paradoxes. I then propose that finer-grained measures of infinite sets from mathematical logic and number theory can help to dissolve the paradoxes in question. Thus, in probability theory, we find that particular kinds of infinitary techniques effectively systematize our theory, while others lead to paradox. (shrink)

D′ ⊆ D is a normal totality on a Scott domain D if it is upward closed and x ⊓ y ∈ D′ is an equivalence relation on D′. We prove that every topological space can be represented by a domain with norma totality.

The spatial organization of the genome is intimately linked to its biological function, yet our understanding of higher order genomic structure is coarse, fragmented and incomplete. In the nucleus of eukaryotic cells, interphase chromosomes occupy distinct.

Théologien, historien, philosophe, sociologue et homme politique libéral, Ernst Troeltsch (1865-1923) fait partie du groupe des théologiens protestants allemands appelé " Ecole de l'histoire des religions ". Revendiquant l'héritage de Kant et de Schleiermacher, proche de Max Weber et du néo-kantisme de l'Ecole de Bade, Troeltsch est le théoricien classique du néo-protestantisme. Surtout connu en France comme sociologue de la religion, il est redécouvert aujourd'hui comme philosophe et théologien, éclipsé un temps par Barth, l'existentialisme et Heidegger. Les huit essais rassemblés (...) ici sont écrits à l'horizon d'une réflexion sur le religieux marquée par les sciences humaines et sociales et par une conscience accusée de la pluralité des formes de la religion. Troeltsch y dessine un programme qui inscrit délibérément la théologie dans une perspective historique, tout en s'efforçant d'articuler approche empirique et interrogation touchant les jugements de valeur. Il récuse ainsi toute tentative pour faire valoir un domaine théologique réservé : non seulement le christianisme n'est pas la religion dans sa forme achevée ou " absolue ", mais les tentatives pour déterminer une " essence " du christianisme par-delà ses inscriptions socioculturelles diverses devront également être refusées. En désenclavant la réflexion théologique, Troeltsch oblige à redéfinir son objet et son statut : elle ne sera plus explicitation du croire, mais devra esquisser une théorie du christianisme précisant le statut, la fonction et les limites du religieux dans une modernité marquée par la diversification des instances de rationalisation et de légitimité. Ce faisant, son propos tranche avec les discours théologiques dominants dans la seconde moitié du XXe siècle, tant en protestantisme qu'en catholicisme. Et il ouvre à nouveau la question d'une pertinence sociale et culturelle du religieux et du théologique. A l'heure où les paradigmes modernes se trouvent en profonde interrogation sur eux-mêmes et où la question religieuse fait retour - peut-être pour le meilleur mais souvent pour le pire -, la lecture de Troeltsch peut apporter une contribution bienvenue à un ensemble de questions parmi les plus centrales de notre temps. Publiés pour la première fois en français, les textes présentés ici sont accompagnés d'un important appareil de commentaires historiques aidant le lecteur à reconstituer le contexte et la genèse des débats en cause. (shrink)

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [9] of P. Resende. As a motivating example, we describe the connection between the quantale and the C*-algebra that both classify Penrose tilings, which was left as an open problem in [5].

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which (...) the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions. (shrink)

A topological index can be focused on uprising of a chemical structure into a real number. The degree-based topological indices have an active place among all topological indices. These topological descriptors intentionally associate certain physicochemical assets of the corresponding chemical compounds. Graph theory plays a very useful role in such type of research directions. The hex-derived networks have vast applications in computer science, physical sciences, and medical science, and these networks are constructed by hexagonal mesh networks. In this paper, we (...) determined the exact values of vertex-edge-degree-based topological descriptors for hex-derived networks HDN 1 p and HDN 2 p, which are generated by the hexagonal network of dimension p. (shrink)

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the (...) Pudlák–Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor generating p-points which are k-arrow but not \-arrow, and in a partial order of Blass producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra \\). If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly \. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template. (shrink)

Our reality constitutes itself as being one of pictures. Landscape is a product of aesthetic reflection as well as the perception of reality and virtual reality of the first order (VR 1). Pictorial representation of a landscape is virtual reality of the second order (VR 2). A picture is a structure of relations with a specific topology or an interrelationship. A picture is set in relation. Topology relates to relational similarities and differences as well as their transfer into other interrelationships, (...) other topologies. The differences between nature, landscape (VR 1), metaphor of landscape and pictorial representation of landscape, etc. (VR 2) describe a change of the interrelationship. These changes happen in a physical-psychic-mental production of reality. We are involved in the events of particular relationships of the picture. The Topology of Being and Appearance reflects the inconspicuous similarities of relations and proportions in the relationship of the world in association with our physical-psychic-mental existential orientation in the world. (shrink)

A three-variable biochemical prototype involving two enzymes with autocatalytic regulation proposed by Decroly and Goldbeter (1987) is analyzed using a topological approach. A two-branched manifold, a so-called template, is thus identified. For certain control parameter values, this template is a horseshoe template with a global torsion of two half-turns. This implies that the bifurcation diagram can be described using the usual sequences associated with a unimodal map with a differentiable maximum as well as exemplified by the logistic map. Moreover, (...) a type-I intermittency associated with a saddle-node bifurcation is exhibited. The dynamics from a single time series are also investigated to determine whether it is possible to investigate the dynamics of this biochemical model from the measure of a single concentration. (shrink)

Topological indices are quantitative measurements that describe a molecule’s topology and are quantified from the molecule’s graphical representation. The significance of topological indices is linked to their use in QSPR/QSAR modelling as descriptors. Mathematical associations between a particular molecular or biological activity and one or several biochemical and/or molecular structural features are QSPRs and QSARs. In this paper, we give explicit expressions of two recently defined novel ev-degree- and ve-degree-based topological indices of two classes of benzenoid, namely, linear hexagonal chain (...) and hammer-like benzenoid. (shrink)

Shehtman introduced bimodal logics of the products of Kripke frames, thereby introducing frame products of unimodal logics. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalize this idea to the bimodal logics of the products of topological spaces, thereby introducing topological products of unimodal logics. In particular, they show that the topological product of S4 and S4 is S4 ⊗ S4, i.e., the fusion of S4 and S4: this logic is strictly weaker than the frame product S4 × S4. In this (...) paper, we axiomatize the topological product of S4 and S5, which is strictly between S4 ⊗ S5 and S4 × S5. We also apply our techniques to (1) proving a conjecture of van Benthem et al concerning the logic of products of Alexandrov spaces with arbitrary topological spaces; and (2) solving a problem in quantified modal logic: in particular, it is known that standard quantified S4 without identity, QS4, is complete in Kripke semantics with expanding domains; we show that QS4 is complete not only in topological semantics with constant domains (which was already shown by Rasiowa and Sikorski), but wrt the topological space Q with a constant countable domain. (shrink)

The qualitative nature of co-deployed or “associated” metaphorical source domains in discourse has been extensively researched, often in terms of whether they share common conceptual roots. There are however few empirical studies on the strength of these associations and their implications. In mental healthcare activities like psychological interviews and counseling, for example, strongly associated sources may suggest unique “conceptual assemblages” that highlight underexplored (dis)similarities between clients or client groups, beyond the typical focus on isolated sources and frequencies. Our (...) case study compares source domain associations produced by two interviewee groups – one meeting the diagnostic threshold for acute stress disorder (ASD), the other not – relating their first-hand experiences of traumatic events during the 2019 Hong Kong social unrest. The machine learning method of association rule mining is used to extract and quantify the top association rules in both groups. Results show i) a shared rule suggesting a highly schematic construal of trauma that nevertheless varies in instantiating details, and ii) several distinct rules that corroborates and lends further insight into the “agentive vs. non-agentive” characteristics exhibited by ASD vs. non-ASD individuals. Implications and future directions, including potential extensions to other discourse contexts, are discussed. (shrink)

We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies. The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, (...) Weinheim. (shrink)

Violence, as a concept, has shaped most of human history and discourse. Over the centuries, the concept has gone through dynamic evolutions and should be understood in relation to diverse agents such as nation, nostalgia, and culture. Modern society’s tendency to impede and constrain overt forms of violence has paved the way for covert forms to exist in socio-cultural spheres. Cultural violence is one such realization where aggression gets exercised covertly through heterogenous mediums such as language, regulations, mass media, and (...) most importantly cultural practices. Its topological structures can be traced in national imagination and a sense of cultural nostalgia originating out of it, that ultimately formulates cultural “otherness.” In Gandhian philosophy, the absence of physical aggression is insignificant, if not complemented with the eradication of violence from the cultural and intellectual strata. Gandhi’s critique of exclusive nationalism and narrowness is reflective of a distinct kind of cultural topology that generates structural violence and with the due course of history it gets legitimacy to exert power over the cultural binary it constructed. The fundamental questions of the paper are associated with assessing the role of national imagination and cultural imperatives in germinating the structures of violence in culture, exclusive nationalism, and Gandhian reconsideration of peace in the context of covert violence in the material and intellectual realms. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( \sqsubseteq (...) \) defined by $a \sqsubseteq b$ if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation $\sqsubseteq$. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

We investigate several ideal versions of the pseudointersection number $\mathfrak {p}$, ideal slalom numbers, and associated topological spaces with the focus on selection principles. However, it turns out that well-known pseudointersection invariant $\mathtt {cov}^*({\mathcal I})$ has a crucial influence on the studied notions. For an invariant $\mathfrak {p}_\mathrm {K}({\mathcal J})$ introduced by Borodulin-Nadzieja and Farkas (Arch. Math. Logic 51:187–202, 2012), and an invariant $\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J})$ introduced by Repický (Real Anal. Exchange 46:367–394, 2021), we have $$\begin{aligned} \min (...) \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathtt {cov}^*({\mathcal I})\}=\mathfrak {p},\qquad \min \{\mathfrak {p}_\mathrm {K}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}\le \mathtt {cov}^*({\mathcal I}), \end{aligned}$$ respectively. In addition to the first inequality, for a slalom invariant \(\mathfrak {sl_e}({\mathcal I},{\mathcal J})\) introduced in Šupina (J. Math. Anal. Appl. 434:477–491, 2016), we show that $$\begin{aligned} \min \{\mathfrak {p}_\mathrm {K}({\mathcal I}),\mathfrak {sl_e}({\mathcal I},{\mathcal J}),\mathtt {cov}^*({\mathcal J})\}=\mathfrak {p}. \end{aligned}$$ Finally, we obtain a consistency that ideal versions of the Fréchet–Urysohn property and the strictly Fréchet–Urysohn property are distinguished. (shrink)

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of [Formula: see text]-indescribability and [Formula: see text]-indescribability of a cardinal [Formula: see text] for all [Formula: see text]. In this context, universal [Formula: see text] formulas exist, there is a normal ideal associated to [Formula: see text]-indescribability and the notions of (...) [Formula: see text]-indescribability yield a strict hierarchy below a subtle cardinal. Additionally, given a regular cardinal [Formula: see text], we introduce a diagonal version of Cantor’s derivative operator and use it to extend Bagaria’s [Derived topologies on ordinals and stationary reflection, Trans. Amer. Math. Soc. 371(3) (2019) 1981–2002] sequence [Formula: see text] of derived topologies on [Formula: see text] to [Formula: see text]. Finally, we prove that for all [Formula: see text], if there is a stationary set of [Formula: see text] that have a high enough degree of indescribability, then there are stationarily many [Formula: see text] that are nonisolated points in the space [Formula: see text]. (shrink)

To generalize as in [7] the constructions of [2] and [5], let L be a nondegenerate join-semilattice with unit. With A · B = {x ∨ y ∈ L : x ∈ A, y ∈ B} and A⊥ = {y ∈ L : x ∨ y = 1 for all x ∈ A}, the structure , ·,∪, ⊥ , L, ∅) is the algebra of subsets for L. Let R be the maximal ideals of L. Interpreting L as the totality (...) of elementary situations, and 1 ∈ L as the impossible one , the members of R will be called realizations, and R itself – the logical space associated with L. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( \sqsubseteq (...) \) defined by $a \sqsubseteq b$ if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation $\sqsubseteq$. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

In this pedagogical communication after demonstrating the legitimacy for using the quantum theory of atoms in molecules (QTAIM) to non-Coulombic systems, Hookean H2 +/H3 2+ species are used for AIM analysis. In these systems, in contrast to their Coulombic counterparts, electron density is atom-like and instead of expected two/three topological atoms, just a single topological atom emerges. This observation is used to demonstrate that what is really “seen” by the topological analysis of electron densities is the clustering of electrons. The (...) very trait of monotonic decay of electron density around the “centers” of clustering guarantees the appearance of topological atoms as basin of attraction of the gradient vector field of the electron density. Although observations with Hookean molecules may seem disappointing at first glance, a careful reasoning points to the fact that the QTAIM methodology is extendable to novel domains, by a knowledge of the morphology of underlying densities, beyond the typical Coulombic systems. (shrink)