It is shown that axially symmetric inhomogeneous magnetic fields can lead to stabilization of higher-order magnetic skyrmions with topological charge |Q| > 1 due to orbital effects. We describe the analytical theory on the energy, size, and domain wall width of the such skyrmions in a power-law inhomogeneous fields, with a parameters corresponding to strongly correlated electron systems. The results obtained may have applications in describing the formation of nontrivial magnetic structures in inhomogeneous fields of superconducting vortices in heterostructures superconductor - chiral magnet of the type [Ir1Fe0.5Co0.5Pt1]10 /MgO/Nb.

A coordinate system of the 2nd kind is being built, corresponding to canonical coordinates of the 1st kind (terminology A. I. Maltsev), thereby obtaining a parametric solution of the system Lie's equations. The integral representation of the group operations f(x, y) of the local Lie group G in canonical coordinates 1st kind. As the main apparatus is used modified formula of A. P. Yuzhakov for implicit mappings. The operation f(x, y) is also represented as a power series, which is the reduced form of the series Campbell-Hausdorff.

The search for analytical profiles of chiral magnetic structures such as 2D magnetic skyrmions (MS) is important for their theoretical study. Since the Euler-Lagrange (EL) equations for such excitations are not solved exactly, the MSs are described using analytical ansatzs. In this work, we validate one of the widely used ansatzs based on a symmetry analysis of the 1D analog of the EL equations, which characterizes the radial profile of the MS. As a development of this approach, a profiles of skyrmion bags are proposed.

We consider discriminant Δ _n of a general polynomial of degree n and Newton polytope 𝒩 for the discriminant, and prove that Newton polytopes with truncations of Δ _n with respect to faces 𝒩 are the Minkowski sums of Newton polytopes for the discriminants of the polynomials of lower degrees.

It is shown that axially symmetric inhomogeneous magnetic fields can lead to stabilization of higher-order magnetic skyrmions with topological charge |Q|>1 due to orbital effects. We developed the analytical theory on the energy, size, and domain wall width of the such skyrmions in a power-law inhomogeneous fields, with a parameters corresponding to strongly correlated electron systems. The results obtained may have applications in describing the formation of nontrivial magnetic structures in inhomogeneous fields of superconducting vortices in heterostructures superconductor --- chiral magnet of the type [Ir1Fe0.5Co0.5Pt1]10/MgO/Nb. Keywords: magnetic skyrmions, nonuniform magnetic fields.

A general polynomial in one variable is considered and the explicit factorization formulas for the truncations of the discriminant with respect to coordinate faces of the polynomial Newton polytope are presented. As a result, the extension of the formulas presented by Gelfand–Kapranov–Zelevinsky is obtained

In this paper we consider the reductant of the dihedral group Dn, consisting of a set of axial symmetries, and the sphere S2 as a reductant of the group SU(2,C) ∼= S3 (the group of unit quaternions). By introducing the Sabinin’s multiplication on the reductant of Dn, we get a quasigroup with unit

The Horn–Kapranov parametrizations describe the singular sets of hypergeometric functions in several variables. These parametrizations are inverses of logarithmic Gauss maps for A-discriminants. In this paper we demonstrate that, despite the multivalued nature of the indicated parametrizations, their blow-ups properties are the same as for single-valued meromorphic mappings. As an application, a new proof of factorization identities for the classical discriminant is given.

Let Δn be the discriminant of a general polynomial of degree n and $$\mathcal{N}$$ be the Newton polytope of Δn. We give a geometric proof of the fact that the truncations of Δn to faces of $$\mathcal{N}$$ are equal to products of discriminants of lesser n degrees. The proof is based on the blow-up property of the logarithmic Gauss map for the zero set of Δn.

The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations.From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term.

We investigate algebraic properties of weakly commutative triples, appearing in the theory of integrable nonlinear partial differential equations. Algebraic technique of skew fields of formal pseudodifferential operators as well as skew Ore fields of fractions are applied to this problem, relating weakly commutative triples to commuting elements of skew Ore field of formal fractions of ordinary differential operators. A version of Burchnall-Chaundy theorem for weakly commutative triples is proved by algebraic means avoiding analytical complications typical for its proofs known in the theory of integrable equations.

A system of n algebraic equations for n unknowns is considered, in which the collection of exponents is fixed, and the coefficients are variable. Since the solutions of such systems are 2n-homogeneous, two coefficients in each equation can be fixed, which makes it possible to pass to the corresponding reduced systems. For the reduced systems, a formula for the solution (and also for any monomial of the solution) is obtained in the form of a hypergeometric type series in the coefficients. Such series are represented as a finite sum of Horn's hypergeometric series: the ratios of the neighboring coefficients of the latter series are rational functions of summation variables. The study is based on the linearization procedure and on the theory of multidimensional residues. As an application of the main formula, a multidimensional analog is presented of the Waring formula for powers of the roots of the system.