ARNOLD  MATHEMATICAL  JOURNAL
Editor-in-Chief:
     Sergei Tabachnikov
Managing Editor:
       Maxim Arnold

A  journal  of    
Published by

Recent Papers

  1. Arithmetic on $q$-deformed rational numbers
    Takeyoshi Kogiso, Kengo Miyamoto, Xin Ren, Michihisa Wakui and Kohji Yanagawa
    Arnold Mathematical Journal, Volume 11, 2025
    Received: 28 October 2024, Accepted: 21 November 2024.
    Abstract
    Recently, Morier-Genoud and Ovsienko introduced a $q$-{deformation} of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal R_{\frac{r}s}(q), ~\mathcal S_{\frac{r}s}(q) \in \mathbb Z[q]$ with $\mathcal R_{\frac{r}s}(1)=r, \mathcal S_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal S_{\frac{r}s}(q)=\mathcal S_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal S_{\frac{r}s}(q)=\mathcal S_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal S_{\frac{r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{-3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.
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  2. Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
    Vladislav Galkin, Olga Pochinka and Danila Shubin
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: 10 March 2024; Accepted: 15 October 2024.
    DOI: 10.56994/ARMJ.011.001.006
    Abstract
    Topological equivalence of Morse-Smale flows without fixed points (NMS-flows) under assumptions of different generalities was studied in a number of papers. In some cases when the number of periodic orbits is small, it is possible to give exhaustive classification, namely to provide the list of all manifolds that admit flows of considered class, find complete invariant for topological equivalence and introduce each equivalence class with some representative flow. This work continues the series of such articles. We consider the class of NMS-flows with unique saddle orbit, under the assumption that it is twisted, on closed orientable 4-manifolds and prove that the only 4-manifold admitting the considered flows is the manifold $\mathbb S^3\times\mathbb S^1$. Also, it is established that such flows are split into exactly eight equivalence classes and construction of a representative for each equivalence class is provided.
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  3. A simple construction of the field of Witt vectors
    Vladimir Fock
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: March 10 2024; Accepted: October 10 2024.
    DOI: 10.56994/ARMJ.011.001.001
    Abstract
    We present a short, hopefully pedagogical construction of the field and ring of Witt vectors. It uses a natural binary operation on polynomials of one variable, which we call convolution.
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  4. Contact geometry of Hill's approximation in a spatial restricted four-body problem
    Cengiz Aydin
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received: 10 March 2024; Accepted: 16 October 2024.
    DOI: 10.56994/ARMJ.011.001.005
    Abstract
    It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-García and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can ``contact'' the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.
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  5. Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards
    Airi Takeuchi and Lei Zhao
    Arnold Mathematical Journal, Volume 11, 2025
    Received: 28 October 2024; Accepted: 21 November 2024.
    Abstract
    The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel transformation. In the first part of this paper, we study how certain integrable mechanical billiards are related by this transformation. This in part illustrates the rotation-invariance of integrable reflection walls in the three-dimensional Kepler billiards found so far. The second part of this paper deals with the Birkhoff-Waldvogel Transformation of the three-dimensional problem wiht two Kepler centers. In particular, we establish an analogous theory of Levi-Civita planes for the Birkhoff-Waldvogel Transformation and show the integrability of certain three-dimensional two-center billiards via a different approach.
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  6. On intersection of lemniscates of rational functions
    Stepan Orevkov and Fedor Pakovich
    Arnold Mathematical Journal, Volume 11, issue 1, 2025
    Received: 10 March 2024; Accepted 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.002
    Abstract
    For a non-constant complex rational function $P$, the {\it lemniscate} of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve given by the equation $L_P(x,y)=0$, where $L_P(x,y)$ is the numerator of the rational function $P(x+iy)\overline{ P}(x-iy)-1.$ In this paper, we study the following two questions: under what conditions two lemniscates have a common component, and under what conditions the algebraic curve $L_P(x,y)=0$ is irreducible. In particular, we provide a sharp bound for the number of complex solutions of the system $\vert P_1(z)\vert =\vert P_2(z)\vert =1$, where $P_1$ and $P_2$ are rational functions.
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  7. Gravitational billiard - bouncing in a paraboloid cavity
    Daniel Jaud
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received 10 March 2024. Accepted: 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.004
    Abstract
    In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of a common sphere of radius $R$. The main results are illustrated in various limiting cases and are compared to its one-dimensional counterpart. In the maximum angular momentum configuration we explicitly state the coordinates of the particle at any time within the cavity.
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  8. Hodge theory on tropical curves
    Yury V. Eliyashev
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025
    Received 10 January 2024. Accepted 10 September 2024.
    DOI: 10.56994/ARMJ.011.001.003
    Abstract
    We construct an analog of the Hodge theory on complex manifolds in the case of tropical curves. We use the analytical approach to the problem, it is based on language of tropical differential forms and methods of $L^2-$cohomologies. In particular, the cohomology groups of a tropical curve can be defined via the de Rham complex of tropical differential forms. We translate standard notions of the complex Hodge theory: the Kähler form, the Hodge star operator, the Laplace-Beltrami operator to the tropical case. The main result of the article is that the tropical Laplace-Beltrami operator is a self-adjoint unbounded operator and the cohomology groups of a tropical curve are isomorphic to the spaces of harmonic forms on this curve.
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  9. Non-fillability of overtwisted contact manifolds via polyfolds
    Wolfgang Schmaltz, Stefan Suhr and Kai Zehmisch
    Arnold Mathematical Journal, Volume 11, 2025
    Received: 3 October 2024. Accepted: 15 October 2024.
    Abstract
    We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive boundary satisfies the weak-filling condition and is overtwisted. Similar results are obtained in the presence of bordered Legendrian open books whose binding–complement has vanishing second Stiefel–Whitney class. The results are obtained via polyfolds.
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Current Issue 11:1 (2025)



Pdf of the whole issue 11:1

Articles of issue 11:1

  1. Vladimir Fock
    A simple construction of the field of Witt vectors
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 1-6.
    Received: March 10 2024; Accepted: October 10 2024.
    DOI: 10.56994/ARMJ.011.001.001
    Abstract
    We present a short, hopefully pedagogical construction of the field and ring of Witt vectors. It uses a natural binary operation on polynomials of one variable, which we call convolution.
    PDF,
    HTML (VersoTeX)

  2. Stepan Orevkov and Fedor Pakovich
    On intersection of lemniscates of rational functions
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 7-26.
    Received: 10 March 2024; Accepted 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.002
    Abstract
    For a non-constant complex rational function $P$, the {\it lemniscate} of $P$ is defined as the set of points $z\in \mathbb C$ such that $\vert P(z)\vert =1$. The lemniscate of $P$ coincides with the set of real points of the algebraic curve given by the equation $L_P(x,y)=0$, where $L_P(x,y)$ is the numerator of the rational function $P(x+iy)\overline{ P}(x-iy)-1.$ In this paper, we study the following two questions: under what conditions two lemniscates have a common component, and under what conditions the algebraic curve $L_P(x,y)=0$ is irreducible. In particular, we provide a sharp bound for the number of complex solutions of the system $\vert P_1(z)\vert =\vert P_2(z)\vert =1$, where $P_1$ and $P_2$ are rational functions.
    PDF,
    HTML (VersoTeX)

  3. Yury V. Eliyashev
    Hodge theory on tropical curves
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 27-77.
    Received 10 January 2024. Accepted 10 September 2024.
    DOI: 10.56994/ARMJ.011.001.003
    Abstract
    We construct an analog of the Hodge theory on complex manifolds in the case of tropical curves. We use the analytical approach to the problem, it is based on language of tropical differential forms and methods of $L^2-$cohomologies. In particular, the cohomology groups of a tropical curve can be defined via the de Rham complex of tropical differential forms. We translate standard notions of the complex Hodge theory: the Kähler form, the Hodge star operator, the Laplace-Beltrami operator to the tropical case. The main result of the article is that the tropical Laplace-Beltrami operator is a self-adjoint unbounded operator and the cohomology groups of a tropical curve are isomorphic to the spaces of harmonic forms on this curve.
    PDF.
    HTML (VersoTeX)

  4. Daniel Jaud
    Gravitational billiard - bouncing in a paraboloid cavity
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 78-97.
    Received 10 March 2024. Accepted: 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.004
    Abstract
    In this work the confined domains for a point-like particle propagating within the boundary of an ideally reflecting paraboloid mirror are derived. Thereby it is proven that all consecutive flight parabola foci points lie on the surface of a common sphere of radius $R$. The main results are illustrated in various limiting cases and are compared to its one-dimensional counterpart. In the maximum angular momentum configuration we explicitly state the coordinates of the particle at any time within the cavity.
    PDF.
    HTML (VersoTeX)

  5. Cengiz Aydin
    Contact geometry of Hill's approximation in a spatial restricted four-body problem
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 98-126.
    Received: 10 March 2024; Accepted: 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.005
    Abstract
    It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-García and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can ``contact'' the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.
    PDF.
    HTML (VersoTeX)

  6. Vladislav Galkin, Olga Pochinka and Danila Shubin
    Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
    Arnold Mathematical Journal, Volume 11, Issue 1, 2025, 127-147.
    Received: 10 March 2024; Accepted: 10 October 2024.
    DOI: 10.56994/ARMJ.011.001.006
    Abstract
    Topological equivalence of Morse-Smale flows without fixed points (NMS-flows) under assumptions of different generalities was studied in a number of papers. In some cases when the number of periodic orbits is small, it is possible to give exhaustive classification, namely to provide the list of all manifolds that admit flows of considered class, find complete invariant for topological equivalence and introduce each equivalence class with some representative flow. This work continues the series of such articles. We consider the class of NMS-flows with unique saddle orbit, under the assumption that it is twisted, on closed orientable 4-manifolds and prove that the only 4-manifold admitting the considered flows is the manifold $\mathbb S^3\times\mathbb S^1$. Also, it is established that such flows are split into exactly eight equivalence classes and construction of a representative for each equivalence class is provided.
    PDF.
    HTML (VersoTeX)

Journal Description

This journal intends to present mathematics so that it would be understandable and interesting to mathematicians independently on their narrow research fields. We invite articles exercising all formal and informal approaches to "unhide" the process of mathematical discovery.

The name of the journal is not only a dedication to the memory of Vladimir Igorevich Arnold (1937-2010), one of the most influential mathematicians of the twentieth century, but also a declaration that the journal hopes to maintain and promote the style which makes the best mathematical works by Arnold so enjoyable and which Arnold implemented in the journals where he was an editor-in-chief.

The ArMJ is run jointly by the Institute for Mathematical Sciences (IMS) at Stony Brook, USA, and The Association for Mathematical Research.


Objectives

The journal intends to publish interesting and understandable results in all areas of Mathematics. The following are the most desirable features of publications that will serve as selection criteria:

  • Accessibility

    The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions that are necessary for understanding must be provided but also informal motivations even if they are well-known to the experts in the field. If a general statement is given, then the simplest examples of it are also welcome.

  • Interdisciplinary and multidisciplinary mathematics

    We would like to have many research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, except for the most popular combinations such as algebraic geometry and mathematical physics, analysis and dynamical systems, algebra and combinatorics, and the like. For this reason, this kind of research is often under-represented in specialized mathematical journals. The ArMJ will try to compensate for this.

  • Problems, objectives, work in progress

    Most scholarly publications present results of a research project in their "final" form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned but the very process of mathematical discovery remains hidden. Following Arnold, we will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. The journal intends to publish well-motivated research problems on a rather regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold's principle, a general formulation is less desirable than the simplest partial case that is still unknown.

  • Being interesting

    The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author's responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author's understanding of the overall picture is presented; however, these parts must be clearly indicated. Including motivations, informal parts, descriptions of other lines of research, possibly conducted by other mathematicians, should serve this principal objective: being interesting.

1.  Types of Journal Articles

  • Research contribution.

    This is the classical format: a short (usually up to 20 pages) account of a research project containing original results and complete proofs of them. However, all of the above applies. Contributions containing very technical arguments may not be suitable for the ArMJ.

  • Research exposition.

    This is an exposition of a broad mathematical subject containing a description of recent results (proofs may be included or omitted), historical overview, motivations, open problems. A research exposition may take 60 pages or more.

  • Problem contribution.

    This is a description of an open problem. The problem must be well-motivated, illustrated by examples, and the importance of the problem must be explained. Alternatively, and closer to the original style of Arnold, a problem contribution may consist of a set of several problems that take very short space to state. Problems do not need to be original, however, the authorship must be carefully acknowledged. A problem contribution is meant to be short (normally, up to 4 pages, but exceptions are possible).

2.  Why the Name

There are many great mathematicians of the twentieth century. The choice of the name may look random (why not, say, "Gelfand Mathematical Journal"? - we are often asked) but we have very specific reasons for using the name of Vladimir Arnold.

  1. The principles, according to which the journal operates, are most accurately associated with Vladimir Arnold. He had been actively promoting these or similar principles.

  2. For many years, V. Arnold had been the Editor-in-Chief of the journal Functional Analysis and its Applications (FAA). In 2006, V. Arnold launched a new journal, Functional Analysis and Other Mathematics (FAOM). The initial composition of the ArMJ Editorial Board consists mostly of former editors of the FAOM.

  3. Despite the close connections with the FAA and the FAOM, we decided to avoid mentioning "Functional Analysis" in the name of the journal. These names have appeared historically, and have nothing to do with scientific principles of the journals. More than that, the names are even confusing: not all mathematicians could guess that, say, Functional Analysis and its Applications welcomes papers in all areas of mathematics, including algebra and number theory. On the other hand, we wanted to have an indication of these connections in the name of the journal. The name of Vladimir Arnold serves as this indication.

Editors

Editor-in-Chief:

   Sergei Tabachnikov, Pennsylvania State University (USA)
e-mail: sot2@psu.edu

Managing Editor:

   Maxim Arnold, University of Texas at Dallas (USA)
email: Maxim.Arnold@utdallas.edu

Editors:

Andrei Agrachev, International School for Advanced Studies (Italy)
e-mail: agrachevaa@gmail.com

Peter Albers, Heidelberg University (Germany)
e-mail: palbers@mathi.uni-heidelberg.de

Gal Binyamini, The Weizmann Institute of Science (Israel)
e-mail: gal.binyamini@weizmann.ac.il

Gil Bor, Centro de Investigaci\'on en Matem\'aticas (Mexico)
e-mail: gil@cimat.mx

Felix Chernous'ko, Institute for Problems in Mechanics, RAS (Russia)
e-mail: chern@ipmnet.ru

Bertrand Deroin, Cergy Paris Universit\'e (France)
bertrand.deroin@gmail.com

David Eisenbud, University of California, Berkeley (USA)
e-mail: de@msri.org

Uriel Frisch, Observatoire de la Côte d'Azur, Nice (France)
e-mail: uriel@oca.eu; uriel@obs-nice.fr

Dmitry Fuchs, University of California, Davis (USA)
e-mail: fuchs@math.ucdavis.edu

Alexander Gaifullin, Steklov Mathematical Institute, Moscow (Russia)
e-mail: agaif@mi-ras.ru

Victor Goryunov, University of Liverpool (UK)
e-mail: Victor.Goryunov@liverpool.ac.uk

Sabir Gusein-Zade, Moscow State University (Russia)
e-mail: sabirg@list.ru

Yulij Ilyashenko, Higher School of Economics, Moscow (Russia)
e-mail: yulijs@gmail.com

Oleg Karpenkov, University of Liverpool (UK)
e-mail: O.Karpenkov@liverpool.ac.uk

Askold Khovanskii, University of Toronto (Canada)
e-mail: askold@math.toronto.edu

Evgeny Mukhin, IUPI, Indianapolis (USA)
e-mail: emukhin@iupui.edu

Anatoly Neishtadt, Loughborough University (UK)
e-mail: A.Neishtadt@lboro.ac.uk

Evita Nestoridi, Stony Brook University (USA)
e-mail: Evrydiki.Nestoridi@stonybrook.edu

Greta Panova, University of Southern California (USA)
e-mail: gpanova@usc.edu

Dan Romik, University of California, Davis (USA)
e-mail: romik@math.ucdavis.edu

Alexander Shnirelman, Concordia University (Canada)
e-mail: alexander.shnirelman@concordia.ca

Frank Sottile, Texas A&M University (USA)
e-mail: sottile@tamu.edu

Vladlen Timorin, Higher School of Economics, Moscow (Russia)
e-mail: vtimorin@hotmail.com

Alexander Varchenko, University of North Carolina, Chapel Hill (USA)
e-mail: anv@email.unc.edu

Oleg Viro, Stony Brook University (USA)
e-mail: oleg.viro@gmail.com

Michael Yampolsky, University of Toronto
e-mail: yampol@math.toronto.edu


Advisors

Artur Avila, University of Zurich (Switzerland) and IMPA (Brasil)

Etienne Ghys, École normale supérieure de Lyon (France)

Dennis Sullivan, Stony Brook University and Graduate Center, CUNY (USA)


Editorial Council

Sergei Tabachnikov, Pennsylvania State University (Editor-in-Chief)

Maxim Arnold, University of Texas, Dallas (Managing Editor)

Vladlen Timorin, Higher School of Economics, Moscow

Oleg Viro, Stony Brook University (A representative of the IMS)

Sabir Gusein-Zade, Moscow State University

Askold Khovanskii, University of Toronto

Yulij Ilyashenko, Higher School of Economics, Moscow and Cornell University

Alexander Varchenko, University of North Carolina, Chapel Hill



Journal of Institute of Mathematical Sciences, Stony Brook University, NY

Published by Association for Mathematical Research, Davis, CA; Jenkintown PA.

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Archive 2015-2024