### Working paper

## Some applications of Griffiths theorem in theory of Feynman integrals

This edited volume gathers selected, peer-reviewed contributions presented at the fourth International Conference on Differential & Difference Equations Applications (ICDDEA), which was held in Lisbon, Portugal, in July 2019.

First organized in 2011, the ICDDEA conferences bring together mathematicians from various countries in order to promote cooperation in the field, with a particular focus on applications. The book includes studies on boundary value problems; Markov models; time scales; non-linear difference equations; multi-scale modeling; and myriad applications.

Background Replicator systems often arise when evolution is concerned. Mathematical models of population dynamics, game theory, economics and biological and molecular evolution lead to the systems of partial differential equations. Due to the absence of analytical solutions for the vast majority of such problems, approximate solutions obtained via numerical simulation are required. Hence, construction of efficient algorithms for the solution of spatial and time-dependent replicator systems is crucial for understanding the dynamics and properties of evolution. Results We give an overview of existing approaches to numerical simulation of replicator systems arising in various fields. We describe a mathematical model of population dynamics with explicit space in game theory setting with asymmetric conflict and a model of biological evolution in presence of a mutator-gene. Both models lead to nonlinear systems of partial differential equations that we cast to the same general form. Then we describe the numerical method based on finite volume framework to solve the system, and provide some numerical examples that demonstrate the method’s validity. Conclusions We conclude that constructed numerical method is suitable for simulation of replicator systems of general form.

The theory of contrasting structures in singularly perturbed boundary problems for nonlinear parabolic partial differential equations is applied to the research of formation of steady state distributions of power within the nonlinear of "Power-Society" model. The interpretation of the solutions to the equation are presented in terms of applied model. The possibility theorem for the problem of getting the solution having some preassigned properties by means of parametric control is proved.

We investigate the multiquantum vortex states in a type-II superconductor in both 'clean' and 'dirty' regimes defined by impurity scattering rate. Within a quasiclassical approach we calculate self-consistently the order parameter distributions and electronic local density of states (LDOS) profiles. In the clean case we find the low temperature vortex core anomaly predicted analytically by Volovik (1993 *JETP Lett.* 58 455) and obtain the patterns of LDOS distributions. In the dirty regime multiquantum vortices feature a peculiar plateau in the zero energy LDOS profile, which can be considered as an experimental hallmark of multiquantum vortex formation in mesoscopic superconductors.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.