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We introduce a new, probability-level approach to calculations in scalar field particle scattering. The approach involves the implicit summation over final states, which makes causality manifest since retarded propagators emerge naturally.…
Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon…
Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to…
We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also…
Caustics are optical phenomena which occur when a family of rays creates an envelope of divergent intensity. Here we show that caustic surfaces also appear when a real or complex field is mapped to its order parameter manifold. We study…
Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these…
Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with…
A representation of partially spatially coherent and partially polarized stationary electromagnetic fields is given in terms of mutually uncorrelated, transversely shifted, fully coherent and polarized elementary electric-field modes. This…
This paper concerns the local study of analytic constrained differential systems (or impasse systems) of the form $A(x)\dot{x} = F(x)$, $x\in\mathbb{R}^{2}$, where $F$ is a vector field and $A$ is a matrix valued function. Using techniques…
The study of bifurcations of differential-algebraic equations (DAEs) is the topic of interest for many applied sciences, such as electrical engineering, robotics, etc. While some of them were investigated already, the full classification of…
Extreme natural disasters can have devastating effects on both urban and rural areas. In any disaster event, an initial response is the key to rescue within 72 hours and prompt recovery. During the initial stage of disaster response, it is…
The unitary crisis for black holes indicates an apparent need to modify local quantum field theory. This paper explores the idea that quantum mechanics and in particular unitarity are fundamental principles, but at the price of familiar…
Documenting the interplay between slow deformation and seismic ruptures is essential to understand the physics of earthquakes nucleation. However, slow deformation is often difficult to detect and characterize. The most pervasive seismic…
The paper explores the differential inclusion of a special form. It is supposed that the support function of the set in the right-hand side of an inclusion may contain the sum of the maximum and the minimum of the finite number of…
It was recently shown that neural ordinary differential equation models cannot solve fundamental and seemingly straightforward tasks even with high-capacity vector field representations. This paper introduces two other fundamental tasks to…
A criterion to locate tricritical points in phase diagrams is proposed. The criterion is formulated in the framework of the Elementary Catastrophe Theory and encompasses all the existing criteria in that it applies to systems described by a…
We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and nonlinear) PDEs, where the term ``weak solution'' comprises the following settings: (a) Distributional solutions. (b) Solutions in…
Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,\lambda)$, $x \in \R^n$, involving one or several constant real parameters $\lambda$. Viewed as a differential equation for the pair $(x,\lambda)$, we…