We investigate uniform versions of (metric) regularity and strong (metric) regularity on compact subsets of Banach spaces, in particular, along continuous paths. These two properties turn out to play a key role in analyzing path-following schemes for tracking a solution trajectory of a&...

We study the solvability of the Zakharov equation in a bounded domain under homogeneous Dirichlet or Navier boundary conditions. This problem is a consequence of the system of equations derived by Zakharov to model the Langmuir collapse in plasma physics. Assumptions for the existe...

This paper is devoted to the investigation of selected situations when computing projective (and other) equivalences of algebraic varieties can be efficiently solved via finding projective equivalences of finite sets of points on the projective line. In particular, we design a method th...

The change in the gut microbiome and microbial metabolites in a patient suffering from severe and enduring anorexia nervosa (AN) and diagnosed with small intestinal bacterial overgrowth syndrome (SIBO) was investigated. Microbial gut dysbiosis is associated with both AN and SIBO, and th...

In this paper, we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also ...

Thomassen's conjecture states that a cubic graph with sufficiently large cyclic connectivity is hamiltonian. Even the following strong conjecture could hold: A cyclically 7-connected cubic graph is hamiltonian, or it is the Coxeter graph. Assuming the conjecture holds true, to prove the&...

Reciprocal pairs of skew morphisms of cyclic groups are in one-to-one correspondence with isomorphism classes of regular dessins with complete bipartite underlying graphs. In this paper we determine all reciprocal pairs of skew morphisms of the cyclic groups provided that one of them&#x...

A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine ...

An ordered pair of integers (m,n) is called singular if g.c.d.(m,\phi(n))=1=g.c.d.(\phi(m),n), a concept which is relevant to pairwise products of cyclic groups. In this note we show that the number of singular pairs is asymptotic to z(x)^2, where z(x) is a function derived&#x...

Graphons are analytic objects representing limits of convergent sequences of graphs. Lovász and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many subgraph densities, has a simple structure. In particular, one of their conjectures would imply&#x...

The 3‐Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2‐regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.

We discuss bichromatic (two-color) front solutions to the bistable Nagumo lattice differential equation. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous 2-periodic equilibria and hence are not monotonic like the standard monochromatic fronts. We provide explicit...

In this paper we deal with the connection of the pouplar card game dobble with combinatorial structures. We show that the existence of perfect decks of cards is connected to the existence of finite projective planes and systems of ortogonal Latin squares. Next, we use a&#...

We discuss multichromatic front solutions to the bistable Nagumo lattice differential equation. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous n -periodic equilibria and hence are not monotonic like the standard monochromatic fronts. In contrast to the...

We study an initial value problem for explicit and implicit difference advection–diffusion equations on graphs. Problems on both finite and infinite graphs are considered. We analyze the existence and uniqueness of solutions. Interestingly, we show that there exist infinitely many solutions ...

Type 2 diabetes mellitus (T2DM) is a major epidemiological problem. Metformin and vildagliptin are well-established antidiabetic drugs. The aim of the study was to evaluate the changes of plasma metabolic profile induced by a high-fat diet (HFD) and subsequent oral administration of met...