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The unreasonable power of the lifting property in elementary mathematics
arxiv
,
2017.
Gavrilovich M.
We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing that many standard elementary notions of abstract topology can be defined by applying the lifting property to simple morphisms of finite topological spaces. Examples in topology include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms. Examples in algebra include: finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms. We include some speculations on the wider significance of this.
Publication based on the results of:
Gavrilovich M., The De Morgan Gazette 2014 Vol. 5 No. 4 P. 23-32
We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms T0 and T1 in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain). We also offer ...
Added: October 20, 2015
Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups
Gavrilovich M., / arxiv. Series math.GR "arxiv". 2017.
We reformulate the statement of the Feit-Thompson theorem in terms of diagrams in the category of finite groups, namely iterations of the Quillen lifting property with respect to particular morphisms. ...
Added: July 20, 2017
Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups
Gavrilovich M., Archive for Mathematical Logic 2018
We reformulate the statement of the Feit-Thompson theorem in terms of diagrams in the category of finite groups, namely iterations of the Quillen lifting property with respect to particular morphisms. ...
Added: October 19, 2018
Gavrilovich M., / arxiv. Series math.GN "arxiv". 2017.
We observe that many of the separation axioms of topology (including T_0-T_4) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually open) continuous maps of finite spaces (involving up to 4 points) and the real line. ...
Added: July 20, 2017
Polotovskiy G., В кн. : Классическая и современная геометрия. Материалы Международной конференции, посвященной 100-летию со дня рождения Л.С. Атанасяна (Москва, 1–4 ноября 2021 г.). : М. : МПГУ, 2021. С. 125-126.
Очерк истории топологического образования в Нижнем Новгороде ...
Added: October 31, 2021
A. N. Krenke, R. B. Sandlersky, A. S. Baybar et al., Известия РАН. Серия биологическая. 2023 Vol. 50 No. 1 P. S85-S99
Four main models of the appearance of boundaries (in a particular case, integrity), arising from the theory of nonlinear dynamic systems, are considered briefly. On the basis of Kotelnikov’s fundamental sampling theorem and, accordingly, general information theory, the character of a distinguished boundary as a function of the sampling frequency in a spatial series with a ...
Added: December 2, 2022
Gavrilovich M., Bulletin of Symbolic Logic 2018
We introduce a simple formal syntax and use it to rewrite in a concise, uniform and intuitive way several standard denitions in topology which are usually expressed in words. The denitions include compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, closed subsets, and some of the separation axioms. The syntax is of ...
Added: October 19, 2018
St. Petersburg : St. Petersburg University, 2016
Abstract of a talk at the 12th International Conference “Logic Today: Developments and Perspectives” ...
Added: June 5, 2018
Efimov A., / Cornell University. Series math "arxiv.org". 2013.
In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary ...
Added: October 31, 2013
Potapova T. A., Молотков С. Н., Письма в Журнал экспериментальной и теоретической физики 2013 Т. 97 № 6 С. 384-391
The symmetry nature of the appearance of specific surface (edge) states at the boundaries of low_dimension structures with the symmetry of ribbons (borders) invariant with respect to time reversal is discussed. Symmetry reasons for the stability of such states against the elastic scattering from nonmagnetic impurities have been revealed. ...
Added: October 11, 2013
Letellier C., Stankevich N., Rössler O., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2022 Vol. 32 No. 2 Article 2230004
Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety ...
Added: February 24, 2022
Athens : The Hellenic Open University, 2013
The book contains the reports of the member of the congress from the different countres. They consider the idea of the symmety in the science and in the art. ...
Added: January 30, 2014
Romanov A., Starykh V., Heliyon 2020 Vol. 6 No. 7 P. 1-7
In this paper we propose and analyze various approaches to organizing routing in a triple loop circulant topologies as applied to networks-on-chip: static routing based on universal graph search algorithms, such as Dijkstra’s algorithm and a possible implementation using Table routing; algorithms created analytically based on an engineering approach with taking into account the structural ...
Added: July 21, 2020
Antonina A. Arkhipova, Ivanov S., Zhuravitskii S. et al., Nanophotonics 2022 Vol. 11 No. 16 P. 3653-3661
We report the experimental observation of the periodic switching of topological edge states between two dimerized fs-laser written waveguide arrays. Switching occurs due to the overlap of the modal fields of the edge states from topological forbidden gap, when they are simultaneously present in two arrays brought into close proximity. We found that the phenomenon ...
Added: October 17, 2022
Shramov K., / Cornell University. Series arXiv "math". 2019.
We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration. ...
Added: November 19, 2019
Dmitry Tonkonog, Tancer M., / Cornell University. Series math "arxiv.org". 2012.
A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable.
More precisely, ...
Added: February 20, 2013
Stepanov E., Buttazzo G., Pratelli A. et al., Berlin : Springer, 2009
Recently much attention has been devoted to the optimization of transportation networks in a given geographic area. One assumes the distributions of population and of services/workplaces (i.e. the network's sources and sinks) are known, as well as the costs of movement with/without the network, and the cost of constructing/maintaining it. Both the long-term optimization and ...
Added: December 12, 2018
Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.
Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...
Added: February 6, 2013
Dragalina-Chernaya E., / Basic Research Programme. Series HUM "Humanities". 2012. No. 10.
The purpose of this paper is twofold: (1) to clarify Ludwig Wittgenstein’s thesis that colours possess logical structures, focusing on his ‘puzzle proposition’ that “there can be a bluish green but not a reddish green”, (2) to compare model-theoretical and game-theoretical approaches to the colour exclusion problem. What is gained, then, is a new game-theoretical ...
Added: December 19, 2012
Balzin E., Успехи математических наук 2014 Т. 69 № 5(419) С. 159-160
В статье дан обзор части результатов диссертационной работы автора. Речь идет о применении идеи категорного разрешения сингулярностей, которая была активно опробована алгебраическими геометрами для триангулированных категорий, в гомотопической алгебре. В связи с тем, что возникающие тут категории не имеют никакой аддитивной структуры, возникает необходимость в разработке новых методов. В рамках формализма Сигала, который позволяет описывать ...
Added: December 24, 2014
Prokhorov Y., , in : 8th European Congress of Mathematics, Portorož, 20–26 June, 2021. : [б.и.], 2023. P. 413-437.
We survey new results on finite groups of birational transformations of algebraic varieties. ...
Added: November 13, 2023
Raikov A., Pirani M., IEEE Access 2022 Vol. 10 P. 56296-56315
The goal of the paper is to find means for the unification of human-machine duality in collective behavior of people and machines, by conciliating approaches that proceed in opposite directions. The first approach proceeds top-down from non-formalizable, cognitive, uncaused, and chaotic human consciousness towards purposeful and sustainable human-machine interaction. The second approach proceeds bottom-up from ...
Added: July 19, 2022
Aleskerov F. T., Shvydun S., , in : Studies in Computational Intelligence. * 1. Vol. 812: Complex Networks and Their Applications VII.: Springer, 2019. P. 94-103.
We propose a model that evaluates how much a network has changed over time in terms of its structure and a set of central elements. The difference of structure is evaluated in terms of node-to-node influence using known nodes correspondence models. To analyze the changes in nodes centralities we adapt an idea of interval orders ...
Added: December 4, 2018