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The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence…
An autonomous system of ordinary differential equations in the plane with a centre-saddle bifurcation is considered. The influence of time damped perturbations with power-law asymptotics is investigated. The particular solutions tending at…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
Summary: A system of autonomous ordinary differential equations depending on a small parameter is considered such that the unperturbed system has an invariant manifold of periodic solutions that is not normally hyperbolic but is normally…
A classic problem in analysis is to solve nonlinear equations of the form \begin{equation*} F(x)=0, \end{equation*} where $F:D^n\to \mathbb{R}^m$ is a continuous map of the closed unit disk $D^n\subset\mathbb{R}^n$ in $\mathbb{R}^m$. A…
In combinatorial group testing problems Questioner needs to find a special element $x \in [n]$ by testing subsets of $[n]$. Tapolcai et al. introduced a new model, where each element knows the answer for those queries that contain it and…
If the fundamental scale of gravity is of the order of 1 TeV, black holes might be produced at the Large Hadron Collider. This work presents simulations of black holes and other exotic models of physics beyond the Standard Model -…
For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local…
Within a semiclassical framework, we investigate spherically symmetric solutions of the Einstein equations that (i) develop a trapped region within a finite time as measured by distant observers, and (ii) remain sufficiently regular at the…
In this paper we propose an elementary topological approach which unifies and extends various different results concerning fixed points and periodic points for maps defined on sets homeomorphic to rectangles embedded in euclidean spaces. We…
This paper consists in discussing some issues on generic local classification of typical singularities of $2D$ piecewise smooth vector fields when the switching set is an algebraic variety. The main focus is to obtain classification results…
The evolution of the atomic structures of the combinatorial library of Sm-substituted thin film BiFeO3 along the phase transition boundary from the ferroelectric rhombohedral phase to the non-ferroelectric orthorhombic phase is explored…
The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic…
The main idea of this paper is to use the notion of buffered failure probability from probabilistic structural design, to introduce buffered environmental contours. Classical environmental contours are used in structural design in order to…
Pseudo algebraically closed, pseudo real closed, and pseudo $p$-adically closed fields are examples of unstable fields that share many similarities, but have mostly been studied separately. In this text, we propose a unified framework for…
We classify, up to a natural equivalence relation, vector fields of the plane which belong to the kernel of a 1--form. This form can be closed, in which case the vector fields are integrable, or not, in which case the differential of the…
We study the properties of quantum cusp and butterfly catastrophes from an algebraic viewpoint. The analysis employs an interacting boson model Hamiltonian describing quantum phase transitions between specific quadrupole shapes by…
A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The…
A first-principles approach to the unitarity problem for black holes is systematically explored, based on the postulates of 1) quantum mechanics 2) the ability to approximately locally divide quantum gravitational systems into subsystems 3)…
We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up…