This paper first demonstrates the existence and uniqueness of solutions to homogeneous Dirichlet boundary value problems for second-order linear elliptic equations with $$L^2$$ -drifts of negative divergence and positive $$L^1$$ -zero-order terms, based on a functional analytic approach, including weak convergence methods and duality arguments. By improving the previous contraction properties, which may not be effective when the zero-order term is very small, this paper introduces a general $$L^2$$ -“contraction” property for any positive zero-order term, leading to remarkable results regarding $$L^2$$ -stability. These stability results are applicable to $$L^2$$ -error analysis for physics-informed neural networks, and can also be applied to stationary Schrödinger operators with $$L^2$$ -zero-order terms. We emphasize that all the constants arising in the estimates of this paper can be explicitly computed.

In $$L_2(\mathbb{R}^d)$$ , we consider a self-adjoint operator which is the sum of a convolution operator and a potential. With minimal assumptions on the convolution kernel and the potential, we describe the location of its essential spectrum and give sufficient conditions for the existence of infinite series of discrete eigenvalues accumulating at the edges of the essential spectrum. We also discuss the case where a non-empty discrete spectrum appears in gaps of the essential spectrum.

We prove that the number of triangulations of a given polygon into triangles with fixed areas of faces is finite, and that an equidissection is algebraic as long as the vertices of the original polygon have algebraic coordinates.

A triangulation of a circle bundle $$E \xrightarrow{\pi} B$$ is a triangulation of the total space $$E$$ and the base $$B$$ such that the projection $$\pi$$ is a simplicial map. In the paper, we address the following questions. Which circle bundles can be triangulated over a given triangulation of the base? What are the minimal triangulations of a bundle? A complete solution for semisimplicial triangulations was given by N. Mnëv. Our results deal with classical triangulations, i.e., simplicial complexes. We give an exact answer for an infinite family of triangulated spheres (including the boundary of the $$3$$ -simplex, the boundary of the octahedron, the suspension over an $$n$$ -gon, the icosahedron). For the general case, we present a sufficient condition for the existence of a triangulation. Some minimality results follow straightforwadly.

Bohl points of a conditionally periodic motion are defined as the phases such that the integral of a continuous function with zero mean value along the motion is always nonnegative (or nonpositive). Bohl points are known to always exist. This note is devoted to a generalization of this result to the case of uniquely ergodic dynamical systems as well as to almost periodic Bohr functions.

Abstract We investigate birational properties of hypersurfaces of degree $$6$$ in the weighted projective space $$\mathbb{P}(1,1,2,2,3)$$ . In particular, we prove that any such quasi-smooth hypersurface is not rational.

We construct an example of a field and a smooth del Pezzo surface of degree $$2$$ over this field without points such that its automorphism group is isomorphic to $$\mathrm{PSL}_2(\mathbb{F}_7) \times \mathbb{Z}/2\mathbb{Z}$$ , which is the largest possible automorphism group for del Pezzo surfaces of degree $$2$$ over an algebraically closed field of characteristic zero.

A rational function on a real algebraic curve $$C$$ is called separating if it takes real values only at real points. Such a function defines a covering $$\mathbb R C\to\mathbb{RP}^1$$ . Let $$c_1,\dots,c_r$$ be the connected components of $$\mathbb R C$$ . M. Kummer and K. Shaw defined the separating semigroup of $$C$$ as the set of all sequences $$(d_1(f),\dots,d_r(f))$$ where $$f$$ is a separating function, and $$d_i(f)$$ is the degree of the restriction of $$f$$ to $$c_i$$ . In the present paper, we describe the separating semigroups of all genus 4 curves. For the proofs, we consider the canonical embedding of $$C$$ into a quadric $$X$$ in $$\mathbb P^3$$ , and apply Abel’s theorem to 1-forms on $$C$$ obtained as Poincaré residues of certain meromorphic 2-forms.

We study a two-dimensional massive Dirac operator with a singular potential supported on a periodic graph, and examine the self-adjointness and the Fredholmness of the associated unbounded operator.

Using the well known approach developed in the papers of B. Davies and his co-authors, we obtain inequalities for the location of possible complex eigenvalues of non-selfadjoint functional difference operators. When studying the sharpness of the main result, we discovered that complex potentials can create resonances.