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Isotopy Classification of Morse Polynomials of Degree Four on R^2
Moscow Mathematical Journal. 2025. Vol. 25. No. 2. P. 249–299.
Isotopy classification of degree 4 Morse polynomials on the real plane is found
Rarovskii A., Journal of Singularities 2025 Vol. 28 P. 217–233
Based on the classification of quasihomogeneous singularities, any polynomial $f$ defining such a singularity can be decomposed as f = f_\kappa + f_{add}. The polynomial f_\kappa takes a specific form, whereas f_{add} is constrained only by the requirement that the singularity of f should be isolated. The polynomial f_{add} is zero if and only if ...
Added: January 22, 2026
Springer, 2025.
Introduction into the contemporary state of Singularity Theory ...
Added: October 31, 2025
Vassiliev V., Journal of Singularities 2024 Vol. 27 P. 47–67
Spaces of Morse perturbation of real simple and parabolic function singularities are studied, in particular the classification of isotopy classes for simple singularities is completed ...
Added: October 31, 2025
Vassiliev V., Moscow Mathematical Journal 2023 Vol. 23 No. 3 P. 401–432
The isotopy classes of non-discriminant perturbations of parabolic function singularities are listed ...
Added: October 31, 2025
Cham: Springer, 2024.
This is the fifth volume of the Handbook of Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. Singularities are ubiquitous in mathematics and science in general, and singularity theory is a crucible where ...
Added: January 29, 2025
Basalaev A., Ionov A., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2024 Vol. 20 Article 024
Consider the pairs (f,G) with f = f(x_1,...,x_N)$ being a polynomial
defining a quasihomogeneous singularity and G being a subgroup of
SL(N,C), preserving f. In particular, G is not
necessary abelian. Assume further that G contains the grading operator
j_f and f satisfies the Calabi-Yau condition. We prove that the
nonvanishing bigraded pieces of the B-model state space of (f,G)
form ...
Added: March 25, 2024
Артемов Н. М., / Series math "arxiv.org". 2022.
The volume function defined by a domain in Euclidean space Rn is the function on the space of affine hyperplanes equal to volumes cut by these hyperplanes from the domain. The study of these functions originates from the works of Archimedes and Newton and is closely related to the theory of lacunas of hyperbolic partial differential equations.The ...
Added: September 28, 2023
Basalaev A., Takahashi A., International Mathematics Research Notices 2022 Vol. 2022 No. 19 P. 14865–14922
For any triple of positive integers A′=(a′1,a′2,a′3) and c∈C∗, cusp polynomial fA′=xa′11+xa′22+xa′33−c−1x1x2x3 is known to be mirror to Geigle–Lenzing orbifold projective line P1a′1,a′2,a′3. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial fA′ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ...
Added: September 9, 2022
Basalaev A., Ionov A., Journal of Geometry and Physics 2022 Vol. 174 Article 104450
For a polynomial f=x_1^n+…+x_N^n let Gf be the non–abelian maximal group of symmetries of f. This is a group generated by all g in GL(N,C), rescaling and permuting the variables, so that f(x)=f(g x). For any G subgroup in Gf we compute explicitly Hochschild cohomology of the category of G–equivariant matrix factorizations of f. We ...
Added: September 9, 2022
Basalaev A., Ionov A., Theoretical and Mathematical Physics 2021 Vol. 209 No. 2 P. 1491–1506
We study Landau-Ginzburg orbifolds (f,G) with f=xn1+…+xnN and G=S⋉Gd, where S⊆SN and Gd is either the maximal group of scalar symmetries of f or the intersection of the maximal diagonal symmetries of f with SLN(ℂ). We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a ...
Added: November 23, 2021
Zhdanovskiy I., Communications in Algebra 2021 Vol. 49 No. 1 P. 43–57
In this article, we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras, we study homotopes by the methods of Category theory and give description of the so-called well-tempered elements of finite-dimensional associative algebra in algebraic terms. ...
Added: September 3, 2021
Basalaev A., Buryak A., International Mathematics Research Notices 2021 Vol. 2021 No. 7 P. 5460–5491
A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types A and D that gives a solution of the open WDVV equations. For the A-singularity, the resulting solution describes ...
Added: April 21, 2020
Ionov A., Journal of Geometry and Physics 2019 Vol. 140 P. 125–130
We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F_{k,n} = ∑^n_{i=1} x^k_i invariant under the natural S_n-action. ...
Added: November 8, 2019
Basalaev A., Takahashi A., Werner E., Journal of Singularities 2023 Vol. 26 P. 92–127
An important invariant of a polynomial f is its Jacobian algebra defined by its partial derivatives. Let f be invariant with respect to the action of a finite group of diagonal symmetries G. We axiomatically define an orbifold Jacobian Z/2Z-graded algebra for the pair (f,G) and show its existence and uniqueness in the case, when ...
Added: February 26, 2019
Basalaev A., ASIAN JOURNAL OF MATHEMATICS 2023 Vol. 26 No. 1 P. 45–80
We give explicitly in the closed formulae the genus zero primary potentials of the three $6$-dimensional FJRW theories of the simple–elliptic singularity $\tilde{E}_7$ with the non–maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov–Witten theory of the elliptic orbifold $[\mathcal{E} / (\mathbb{Z}/2\mathbb{Z})]$ — the orbifold quotient of ...
Added: February 26, 2019
Basalaev A., Takahashi A., Werner E., Arnold Mathematical Journal 2017 Vol. 3 No. 4 P. 483–498
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-Hübsch transform of an invertible polynomial defining the strange dual singularity in the sense of Arnold. ...
Added: February 26, 2019
Генов П. Г., Тимербаев В. Х., Долгашева Н. С. et al., Вопросы нейрохирургии им. Н.Н. Бурденко 2018 Т. 82 № 3 С. 29–35
The risk factors for acute pain as well as chronic pain syndrome (CPS) in spine surgery have not been defined to date. Purpose — to define the prognostic parameters of acute pain severity and the risk of CPS in patients operated on for spinal diseases and injuries. Material and methods. The study included 291 patients ...
Added: October 2, 2018
Кондакчьян Н. А., Абгарян Н. Б., Медицинское информационное агенство, 2018.
The study of clinical terminology has always occupied a significant place in the discipline "Latin language and outlines of medical terminology." Undoubtedly, surgical terminology is one of the most voluminous terminology in the clinical block. Moreover, there are a lot of terms used in it in other departments of the clinical direction, so-called "common" terms. ...
Added: September 26, 2018
Vassiliev V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 825–836
The local multiplicities of the caustics, the Maxwell sets, and the complex Stokes' sets in the spaces of versal deformations of Pham singularities (that is, of germs of holomorphic functions C^n --> C^1 which are expressed by the sums of degrees of the coordinate functions) are calculated ...
Added: December 27, 2017
Kurenkov E., Рязанова К. А., Динамические системы 2017 Т. 7 № 2 С. 113–118
В настоящей работе рассматриваются, периодические сдвиги на $n$-мерном торе, и для двух топологически сопряженных сдвигов исследуется множество сопрягающих их гомеоморфизмов. Из результатов Я.~Нильсена~\cite{Ni} следует, что для периодических гомеоморфизмов двумерного тора таких, что все точки имеют один период, период является полным инвариантом топологической сопряженности. В настоящей работе исследуется вопрос, когда два периодических сдвига на $n$-мерном торе ...
Added: November 15, 2017
Бурина О. А., В кн.: Способы мысли, пути говорения.: СПб.: Алетейя, 2016. С. 275–299.
This article reviews the archaeography of translations of Felix Platter’s «Tagebuch» into German and French languages, while the original had written on the Basel dialect. 19th century translators passed many details and translated only those paragraphs that they considered interesting. In 1878, the first complete translation of the Felix Platter’s «Tagebuch» had published. Based on ...
Added: December 15, 2016