Arnold Mathematical Journal最新文献

A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. We describe the local structure of Gavrilov flows in terms of the geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for axisymmetric Gavrilov flows and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.

We prove that the space of affine, transversal at infinity, nonsingular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.

We describe supertraces on “queerifications” (see arXiv:2203.06917) of the algebras of matrices of “complex size”, algebras of observables of Calogero–Moser model, Vasiliev higher spin algebras, and (super)algebras of pseudo-differential operators. In the latter case, the supertraces establish complete integrability of the analogs of Euler equations to be written (this is one of several open problems and conjectures offered).

The aim of this paper is to study in details the regular holonomic (D-)module introduced in Barlet (Math Z 302 (n^03): 1627–1655, 2022 arXiv:1911.09347 [math]) whose local solutions outside the polar hyper-surface ({Delta (sigma ).sigma _k = 0 }) are given by the local system generated by the power (lambda ) of the local branches of the multivalued function which is the root of the universal degree k equation (z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0 ). We show that for (lambda in mathbb {C} {setminus } mathbb {Z}) this D-module is the minimal extension of the holomorphic vector bundle with an integrable meromorphic connection with a simple pole which is its restriction on the open set ({sigma _kDelta (sigma ) not = 0}). We then study the structure of these D-modules in the cases where (lambda = 0, 1, -1) which are a little more complicated, but which are sufficient to determine the structure of all these D-modules when (lambda ) is in (mathbb {Z}). As an application we show how these results allow to compute, for instance, the Taylor expansion of the root near (-1) of the equation:

near (z^k - (-1)^k = 0).

For the calculation of Springer numbers (of root systems) of type (B_n) and (D_n), Arnold introduced a signed analogue of alternating permutations, called (beta _n)-snakes, and derived recurrence relations for enumerating the (beta _n)-snakes starting with k. The results are presented in the form of double triangular arrays ((v_{n,k})) of integers, (1le |k|le n). An Arnold family is a sequence of sets of such objects as (beta _n)-snakes that are counted by ((v_{n,k})). As a refinement of Arnold’s result, we give analogous arrays of polynomials, defined by recurrence, for the calculation of the polynomials associated with successive derivatives of (tan x) and (sec x), established by Hoffman. Moreover, we provide some new Arnold families of combinatorial objects that realize the polynomial arrays, which are signed variants of André permutations and Simsun permutations.

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences, some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper, we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

Let (pin {mathbb {Z}}^n) be a primitive vector and (Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min (pcdot z, 0)). The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to (Psi ) “at infinity”. We apply this result to sandpile models on ({mathbb {Z}}^n). We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function

there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with (Psi ) “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

We establish the existence of stationary solutions for certain systems of reaction–diffusion-type equations in the corresponding (H^{2}) spaces. Our method relies on the fixed point theorem when the elliptic problem contains second-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates involved in the equations of the systems.

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.