Abstract:
I present a systematic analysis of formation of the universal annihilation catastrophe which develops in an open system, where species $A$ and $B$ diffuse from the bulk of restricted medium and die on its surface (desorb) by the reaction $A + B \to 0$. This phenomenon arises in the diffusion-controlled limit as a result of self-organizing explosive growth (drop) of the surface concentrations of, respectively, slow and fast particles ({\it concentration explosion}) and manifests itself in the form of an abrupt singular jump of the desorption flux relaxation rate. As striking results I find the dependences of time and amplitude of the catastrophe on the initial particle number, and answer the basic questions of when and how universality is achieved.

Abstract:
We study dynamics of filling of an initially empty finite medium by diffusing particles $A$ and $B$, which arise on the surface upon dissociation of $AB$ molecules, impinging on it with a fixed flux density $I$, and desorb from it by the reaction $A + B\to AB\to 0$. We show that once the bulk diffusivities differ ($p=D_{A}/D_{B}<1$), there exists a critical flux density $I_{c}(p)$, above which the relaxation dynamics to the steady state is qualitatively changed: on time dependencies of $c_{As}/c_{e}$ ($c_{e}$ being the steady state concentration at $t\to \infty$) a maximum appears, the amplitude of which grows both with $I$ and with $D_{B}/D_{A}$ ratio. In the diffusion-controlled limit $I \gg I_{c}$ at $p \ll 1$ the reaction "selects" the {\it universal laws} for the particles number growth ${\cal N}_{A}={\cal N}_{B}\propto t^{1/4}$ and the evolution of the surface concentrations $c_{As}\propto t^{-1/4},c_{Bs}\propto t^{1/4}$, which are approached by one of the {\it two characteristic regimes} with the corresponding hierarchy of the intermediate power-law asymptotics. In the first of these $c_{As}$ goes through a comparatively {\it sharp} max$(c_{As}/c_{e})\propto I^{1/6}$, the amplitude of which is $p$-independent, in the second one $c_{As}$ goes through a {\itplateau-like} max$(c_{As}/c_{e})\propto p^{-1/4}$, the amplitude of which is $I$-independent. We demonstrate that on the main filling stage the evolution of the ${\cal N}(t)/{\cal N}_{e}, c_{As}(t)/c_{e},$ and $c_{Bs}(t)/c_{e}$ trajectories with changing $p$ or $J$ between the limiting regimes is unambiguously defined by the value of the scaling parameter ${\cal K}=p^{3/2}J$ ($J$ being the reduced flux density) and is described by the set of {\it scaling laws}, which we study in detail analytically and numerically.

Abstract:
The aim of this report is to give an insight into the key features of the phenomenon called Annihilation Catastrophe which may pretend to be one of the most dramatic manifestations of the reaction-diffusion interplay.

Abstract:
We consider the diffusion-controlled annihilation dynamics $A+B\to 0$ with equal species diffusivities in the system where an island of particles $A$ is surrounded by the uniform sea of particles $B$. We show that once the initial number of particles in the island is large enough, then at any system's dimensionality $d$ the death of the majority of particles occurs in the {\it universal scaling regime} within which $\approx 4/5$ of the particles die at the island expansion stage and the remaining $\approx 1/5$ at the stage of its subsequent contraction. In the quasistatic approximation the scaling of the reaction zone has been obtained for the cases of mean-field ($d \geq d_{c}$) and fluctuation ($d < d_{c}$) dynamics of the front.

Abstract:
We present the growth dynamics of an island of particles $A$ injected from a localized $A$-source into the sea of particles $B$ and dying in the course of diffusion-controlled annihilation $A+B\to 0$. We show that in the 1d case the island unlimitedly grows at any source strength $\Lambda$, and the dynamics of its growth {\it does not depend} asymptotically on the diffusivity of $B$ particles. In the 3d case the island grows only at $\Lambda > \Lambda_{c}$, achieving asymptotically a stationary state ({\it static island}). In the marginal 2d case the island unlimitedly grows at any $\Lambda$ but at $\Lambda < \Lambda_{*}$ the time of its formation becomes exponentially large. For all the cases the numbers of surviving and dying $A$ particles are calculated, and the scaling of the reaction zone is derived.

Abstract:
We consider the problem of diffusion-controlled evolution of the system $A$-particle island - $B$-particle island at propagation of the sharp annihilation front $A+B\to 0$. We show that this general problem, which includes as particular cases the sea-sea and the island-sea problems, demonstrates rich dynamical behavior from self-accelerating collapse of one of the islands to synchronous exponential relaxation of the both islands. We find a universal asymptotic regime of the sharp front propagation and reveal limits of its applicability for the cases of mean-field and fluctuation fronts.

Abstract:
The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we use the abstract theory of filtered and graded Lie algebras. We also describe all differential and integral invariants of new Lie algebras found in the paper and discuss the infinite-dimensional case.

Abstract:
While every group is isomorphic to a transitive group of permutations, the analogous property fails for inverse semigroups: not all inverse semigroups are isomorphic to transitive inverse semigroups of one-to-one partial transformations of a set. We describe those inverse semigroups that are.