We give a complete answer for the question asked by Develin, Santos, and Sturmfels by proving that every 5×n matrix with tropical rank equal to 3 has Kapranov rank equal to 3, for the ground field that contains at least 4 elements. For the ground field either F2 or F3, we construct an example of a 5 × 5 matrix with tropical rank 3 and Kapranov rank 4.
The asymptotic properties of nonoscillating solutions of Emden-Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the asymptotic properties of solutions with integer-valued asymptotics as well as of solutions arising from the rapid decrease of the coefficient of the equation. To analyze the asymptotic behavior of solutions of the equations, methods of power geometry are used.
We obtain an inequality imvolving Betti numbers of six-dimensional hyperk\"ahler manifolds using Rozansky-Witten invariants described by Hitchin and Sawon.
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations equipped with a cyclic group action.
We pdiscuss a constructive representation of R. Feynman formula for derivation of exonential families of noncommuting operators.
This paper deals with list colorings of uniform hypergraphs. Let H(m, r, k) be the complete r-partite k-uniform hypergraph with parts of equal size m in which each edge contains exactly one vertex from some k ≤ r parts. Using results on multiple covers by independent sets, we establish that, for fixed k and r, the list-chromatic number of H(m, r, k) is (1 + o(1)) logr/(r−k+1)(m) as m → ∞.
V.I.Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs invariant with respect to the action σ (x1; y1, …, yn) = (−x1; y1, …, yn) of the group ℤ2. In particular, it was shown that a function germ (a boundary singularity germ) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in 3 + 4s variables stable equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding subspace is finite. We formulate and prove analogs of these statements for function germs invariant with respect to an arbitrary action of the group ℤ2, and also for corner singularities.
In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. The reducibility of such systems to pseudonormal form is studied.
We consider the space U(T) of all continuous functions on the circle T with uniformely convergent Fourier series. We obtain an estimate for the growth of the U -norms of exponential functions with an arbitrary piecewise linear phase and unboundedly growing integer frequences.
We study the Titchmarsh Q-integral, its generalization, and its elementary properties are studied; integrability criteria on sets of finite measure are obtained.