By an additive action on an algebraic variety of dimension we mean a regular action with an open orbit of the commutative unipotent group . We prove that if a complete toric variety admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety are in bijection with complete collections of Demazure roots of the fan . Moreover, any two normalized additive actions on are isomorphic.

We prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.

The classical A. Markov inequality establishes a relation between the maximum modulus or the L∞ ([−1, 1]) norm of a polynomial Qn and of its derivative: ||Qʹn|| ≤Mnn2||Qn|| where the constant Mn = 1 is sharp. The limiting behavior of the sharp constants Mn for this inequality, considered in the space L2 ([−1, 1], w(α,β)(x) with respect to the classical Jacobi weight w(α,β)(x) := (1−x)α(x+1)β, is studied. We prove that, under the condition |α − β| & 4, the limit is limn→∞Mn = 1/(2jν) where jν is the smallest zero of the Bessel function Jν(x) and 2ν = min(α, β) − 1. © 2015, American Mathematical Society.

An n × n sign pattern S, which is a matrix with entries 0, +, −, is called spectrally arbitrary if any monic real polynomial of degree n can be realized as a characteristic polynomial of a matrix obtained by replacing the nonzero elements of S by numbers of the corresponding signs. A sign pattern S is said to be a superpattern of those matrices that can be obtained from S by replacing some of the nonzero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known Nilpotent Jacobian method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern S and its superpattern S′ such that S is spectrally arbitrary but S′ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche.

We investigate flexibility of affine varieties with an action of a linear algebraic group. Flexibility of a smooth affine variety with only con- stant invertible functions and a locally transitive action of a reductive group is proved. Also we show that a normal affine complexity-zero horospherical variety with only constant invertible functions is flexible.

Answering a question of J. Kovacic, we show that for any Keigher ring, its differential spectrum coincides with the differential spectrum of the ring of global sections of the structure sheaf. In particular, we obtain the answer for Ritt algebras, that is, differential rings containing the rational numbers.

A locally conformally Kähler (LCK) manifold is a complex manifold M admitting a Kähler covering \tilde{M}, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if its covering admits an automorphic Kähler potential. It is known that in this case \tilde{M} is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kähler metrics with potential which could arise from an LCK structure is in bijective correspondence with the set of pseudoconvex shells, that is, pseudoconvex hypersurfaces in C meeting each orbit of the associated \mathbb{R}^{>0}-action exactly once and transversally. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea, Belgun and Kamishima-Ornea.

We show that neither the Barvinok rank nor the Kapranov rank of a tropical matrix M can be defined in terms of the regular mixed subdivision produced by M. This answers a question asked by Develin, Santos and Sturmfels.

In this paper we construct an infinite family of paramodular forms of weight 2 which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen (2013) related to the only known infinite series of theta-blocks of weight 2 and q-order 1. We also consider some applications of this result.

We give a simple proof of the Emch closing theorem by introducing a new invariant measure on the circle. Special cases of that measures are well-known and have been used in the literature to prove Poncelet’s and Zigzag theorems. Some further generalizations are also obtained by applying the new measure.