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A diagram chasing formalisation of elementary topological properties
Gavrilovich M.
We introduce a simple formal syntax and use it to rewrite in a concise, uniform and intuitive way several standard de nitions in topology which are usually expressed in words. The de nitions include compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, closed subsets, and some of the separation axioms. The syntax is of category-theoretic avour and is based on the observation that these properties can all be de ned category-theoretically by repeated application of a standard category theory trick, the Quillen lifting property (orthogonality
of morphisms), starting from a single (counter)example.
We hope our reformulations may be use in formalisation of elementary topology and in teaching.
Priority areas:
mathematics
Language:
English
Keywords: topology
Publication based on the results of:
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Taletskii D., / Series arXiv "math". 2026.
A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...
Added: May 1, 2026
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...
Added: April 18, 2026
Medvedev V., / Series arXiv "math". 2026.
We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a ...
Added: April 3, 2026
Gabdullin N., Androsov I., / Series Computer Science "arxiv.org". 2026.
Label prediction in neural networks (NNs) has O(n) complexity proportional to the number of classes. This holds true for classification using fully connected layers and cosine similarity with some set of class prototypes. In this paper we show that if NN latent space (LS) geometry is known and possesses specific properties, label prediction complexity can ...
Added: April 2, 2026
Lukacheva S., Zakharova E., Makhaneva A. et al., Journal of Solid State Chemistry 2025 Vol. 345 Article 125223
A new gallium-rich intermetallic Ti2-xNi3Ga9 was synthesized by high-temperature techniques as bulk samples and single crystals. The composition and structure of the compound were established by a combination of single-crystal and powder X-ray analysis and energy-dispersive X-ray spectroscopy. The obtained intermetallic compound crystallizes in its own structure type and has a narrow homogeneous region of ...
Added: February 23, 2026
Kolesnikov A., / Series arXiv "math". 2025.
We study Blaschke--Santal{ó}-type inequalities for N>=2 sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of N>2 sets.
We also discuss links to the ...
Added: February 13, 2026
Sorokin K., Beketov M., Онучин А. et al., / arxiv.org. Серия cs.SI "Social and Information Networks ". 2025.
Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the ...
Added: January 15, 2026
Gaianov N., Parusnikova A., / Cornell University. Серия math "arxiv.org". 2025.
An algebraic q-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain q-difference analogue of the fifth Painlevé equation ...
Added: December 25, 2025
Popov V., / Series arXiv "math". 2025. No. 2502.01539.
We prove that the variety of flexes of algebraic curves
of degree 3 in the projective plane is an ideal theoretic complete
intersection in the product of a two-dimensional and a nine-dimensional projective spaces. ...
Added: December 16, 2025
Gnetov F., Konakov V., / Series arXiv "math". 2025. No. 2512.04667.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural
random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which
includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes
the real hyperbolic space Hn. Our approach introduces ...
Added: December 5, 2025
Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.
The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...
Added: December 4, 2025
Bitter I., Konakov V., / Cornell University. Серия arXiv "math". 2025. № 2505.24548.
В работе приводится обобщение локальной предельной теоремы о сходимости неоднородных цепей Маркова к диффузионному пределу на случай, когда соответ- ствующие коэффициенты процессов удовлетворяют слабым условиям регулярности и совпадают лишь асимптотически. В частности, рассматриваемые нами коэффици- енты сноса могут быть неограниченными с не более чем линейным ростом, а оценки отражают перенос терминального состояния неограниченным трендом через ...
Added: December 3, 2025
Bogomolov F. A., Schrandt S., / Series arXiv "math". 2025.
We discuss phenomena of stabilization for direct images of line bundles over projective curves mapping onto the projective line, for maps of sufficiently big degree. ...
Added: December 1, 2025
Deviatov R., Baek S., / Series arXiv "math". 2025.
The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of ...
Added: December 1, 2025
Hessian-based lightweight neural network for brain vessel segmentation on a minimal training dataset
Меньшиков И. А., Бернадотт А. К., Elvimov N. S., / Series arXie "Statistical mechanics". 2025.
Accurate segmentation of blood vessels in brain magnetic resonance angiography (MRA) is essential for successful surgical procedures, such as aneurysm repair or bypass surgery. Currently, annotation is primarily performed through manual segmentation or classical methods, such as the Frangi filter, which often lack sufficient accuracy. Neural networks have emerged as powerful tools for medical image ...
Added: December 1, 2025
Prokhorov Y., / Series arXiv "math". 2025.
A $\mathbf{Q}$-conic bundle is a contraction $f: X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\uprho(X/Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form. ...
Added: December 1, 2025
Amerik E., Verbitsky M., Soldatenkov A., / Series arXiv "math". 2025.
Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction ...
Added: December 1, 2025
Дюканов П. А., Kozhukhov M., Popov D. et al., Успехи прикладной физики 2024 Т. 12 № 4 С. 334–342
The paper considers options for the design of an n-channel junction-gate field-effect transistor, integrated into a high-voltage complementary bipolar technological process with isolation by a reverse-biased p-n junction. Criteria for the selection of transistor models according to electrophysical parameters are formulated. Taking into account the criteria, the instrument-technological modeling on bulk silicon substrates was carried ...
Added: September 12, 2024
Mikhail Yu. Romashikhin, , in: Proceedings 2024 International Russian Smart Industry Conference (SmartIndustryCon), 25-29 March 2024.: Sochi: IEEE, 2024. P. 104–108.
This article describes the implementation of regular topologies for networks-on-chip. The complexity of network development and its main parameters depend on the choice of topology. The rationale for the influence of topology on network bandwidth is given. Software that automates the generation of configuration files (bitstream) for the implementation of networks-on-chip with different topologies and ...
Added: May 13, 2024
Krenke A., Sandlerskiy R., Baibar A. et al., Известия РАН. Серия биологическая 2023 № 7 С. 93–108
A brief review of the main models for emergence of boundaries and, in a particular case, integrity, arising from the theory of nonlinear volumetric systems. Based on the fundamental value of Kotelnikov’s reference and, accordingly, the general theory of awareness about identifying a characteristic border feature, as frequency of sampling functions in incidence along with ...
Added: December 9, 2023
Gogoleva N., Zorina-Tikhonova E., Khapaeva P. et al., Journal of Molecular Structure 2022 Vol. 1256 Article 132532
A series of cobalt(II) and cadmium(II) compounds comprising dimethylmalonate anions combined with potassium and barium ions have been synthesized. The compounds obtained are coordination polymers with 3D structure (1) or 2D structure (2–4). It has been shown that, depending on the nature of heterometals, the type of coordination of dimethylmalonic acid by d-metal atoms changes: from ...
Added: January 24, 2023
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