By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of asymptotic average solutions making pointwise solvable every Poisson equation ${\mathcal {L}} u(x)=-f(x)$ with continuous data f, where {L}}$ is hypoelliptic linear partial differential operator positive semidefinite characteristic form.

— We prove: 1) for any linear language L, it is recursively unsolvable to détermine whether L is an 1RS language; 2) for any deterministic context-free language L, it is recursively solvable to détermine whether L is an 1RS language; 3) there exists an 1RS linear language containing the concaténation product oftwo infinité contextfree languages.

Suppose that the inverse scattering problem is understood as follows: given fixed-energy phase shifts, corresponding to an unknown potential q = q(r) from a certain class, for example, q ∈ L 1,1 , recover this potential. Then it is proved that the Newton-Sabatier (NS) procedure does not solve the above problem. It is not a valid inversion method, in the following sense: 1) it is not possible to...

As far as we know, for most polynomially solvable network optimization problems, their inverse problems under l1 or l∞ norm have been studied, except the inverse maximum-weight matching problem in non-bipartite networks. In this paper we discuss the inverse problem of maximum-weight perfect matching in a non-bipartite network under l1 and l∞ norms. It has been proved that the inverse maximum-we...

Suppose that the inverse scattering problem is understood as follows: given fixed-energy phase shifts, corresponding to an unknown potential q = q(r) from a certain class, for example, q ∈ L 1,1 , recover this potential. Then it is proved that the Newton-Sabatier (NS) procedure does not solve the above problem. It is not a valid inversion method, in the following sense: 1) it is not possible to...

An induced matching in a graph G is a set of edges, no two of which meet a common node or are joined by an edge of G; that is, an induced matching is a matching which forms an induced subgraph. Induced matchings in graph G correspond precisely to independent sets of nodes in the square of the line-graph of G, which we denote by [L(G)]. Often, if G has a nice representation as an intersection gr...

For every integer g, we construct a 2-solvable and 2-bipolar knot whose topological 4-genus is greater than g. Note that knots are in particular algebraically slice have vanishing Casson-Gordon obstructions. Similarly all known smooth bounds from gauge theory Floer homology vanish for knots. Moreover, our bound smoothly embedded height four gropes $D^4$, an priori stronger condition being 2-sol...

The main purpose of this paper is a wide generalization one the results abstract algebraic geometry begins with, namely fact that prime spectrum $${\mathrm {Spec}}(R)$$ unital commutative ring R always spectral (= coherent) topological space. In generalization, which includes several other known ones, role ideals played by elements an complete lattice L equipped with binary multiplication $$xy\...

All finite-dimensional indecomposable solvable Lie algebras L(n, f), having the triangular algebra T (n) as their nilradical, are constructed. The number of nonnilpotent elements f in L(n, f) satisfies 1 ≤ f ≤ n− 1 and the dimension of the Lie algebra is dim L(n, f) = f + 1 2 n(n − 1).

Applications of hypergroups have mainly appeared in special subclasses. One of the important subclasses is the class of polygroups. In this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. In particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.