Let a finite abelian group G act (linearly) on the space R^n and thus on its complexification C^n. Let W be the real part of the quotient C^n/G (in general W \neq R^n/G). We give an algebraic formula for the radial index of a 1-form \omega on the real quotient W. It is shown that this index is equal to the signature of the restriction of the residue pairing to the G-invariant part \Omega^G_\omega of \Omega_\omega=\Omega^n_{R^n,0}/\omega \wedge \Omega^{n-1}_{R^n,0}. For a G-invariant function f, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function f, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form df on the preimage \pi^{-1}(W) of W under the natural quotient map.

We consider fourth order ordinary differential operators on the half-line and on the line, where the perturbation has compactly supported coefficients. The Fredholm determinant for this operator is an analytic function in the whole complex plane without zero.We describe the determinant at zero.We show that in the generic case it has a pole of order 4 in the case of the line and of order 1 in the case of the half-line.

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta-shock solution of the continuity equation for a discontinuous velocity field.We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).

Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite an implicit way, we find explicitly the numbers of the corresponding extremal eigenvalues. In particular we provide an extremal metric for the third eigenvalue of the torus.

The paper introduces a class of functions where the resolving operator for a system of Kolmogorov–Feller-type equations with a small parameter is well posed in forward and backward times. The introduced class of functions is invariant under the resolving operator if the solution is understood in the weak sense with an exponential weight. The paper continues the study of [6].

We prove that if *X* is a rationally connected threefold and *G* is a *p*‐subgroup in the group of birational selfmaps of *X*, then *G* is an abelian group generated by at most 3 elements provided that . We also prove a similar result for under an assumption that *G*acts on a (Gorenstein) *G*‐Fano threefold, and show that the same holds for under an assumption that *G* acts on a *G*‐Mori fiber space.

An irreducible algebraic variety *X* is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of is pointed, the group is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.