Arnold Mathematical Journal最新文献

We investigate the question of how many subgroups of a finite group are not in its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasolă and Tărnăuceanu (Bull Aust Math Soc 107(3):451–455, 2023) asked how many subgroups are not in the Chermak–Delgado lattice and classified all groups with two or less subgroups not in the Chermak–Delgado lattice. We extend their work by classifying all groups with less than five subgroups not in the Chermak–Delgado lattice. In addition, we show that a group with less than five subgroups not in the Chermak–Delgado lattice is nilpotent. In this vein, we also show that the only non-nilpotent group with five or fewer subgroups in the Chermak–Delgado lattice is (S_3).

In this paper, the normality of a family of meromorphic functions is deduced from the normality of a given family. Precisely, we have proved: Let ({mathcal {F}}) and ({mathcal {G}}) be two families of meromorphic functions on a domain D, and (a, b, c) be three finite complex numbers such that (ane 0) and (bne c). Suppose that ({mathcal {G}}) is normal in D such that no sequence in ({mathcal {G}}) converges locally uniformly to infinity in D. If (nge 3) and for each function (fin {mathcal {F}}) there exists (gin {mathcal {G}}) such that (f^{'}-af^{n}) and (g^{'}-ag^{n}) partially share the values b and c, then ({mathcal {F}}) is normal in D. Further, examples are given to establish the sharpness of the result.

We investigate the effect of an (varepsilon )-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric d-tensor f, a norm (leftVert cdot rightVert ) on the space of symmetric d-tensors, and (varepsilon >0) are given. What is the smallest symmetric tensor rank in the (varepsilon )-neighborhood of f? In other words, what is the symmetric tensor rank of f after a clever (varepsilon )-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind, we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.

The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over ({mathbb {C}}) can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).

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.