Acta Applicandae Mathematicae

The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u t =F(t,x,u,u x )u xx +G(t,x,u,u x ). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.

We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$ where $\lambda >0$ is a parameter, $p\in (2,4)$ and $$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$ We prove that the system has no solutions for $\lambda $ large and has two radial solutions for $\lambda $ small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of $\lambda $. Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).

In this paper, we consider dependent random variables X k , k=1,2,… with supports on [−b k ,∞), respectively, where the b k ≥0 are some finite constants. We derive asymptotic results on the tail probabilities of the quantities $S_{n}=\sum_{k=1}^{n} X_{k}$ , X (n)=max 1≤k≤n X k and S (n)=max 1≤k≤n S k , n≥1 in the case where the random variables are dependent with heavy-tailed (subexponential) distributions, which substantially generalize the results of Ko and Tang (J. Appl. Probab. 45, 85–94, 2008).

If (M,g) and $(\hat{M},\hat{g})$ are two smooth connected complete oriented Riemannian manifolds of dimensions n and $\hat{n}$ respectively, we model the rolling of (M,g) onto $(\hat{M},\hat{g})$ as a driftless control affine systems describing two possible constraints of motion: the first rolling motion (Σ) NS captures the no-spinning condition only and the second rolling motion (Σ) R corresponds to rolling without spinning nor slipping. Two distributions of dimensions $(n + \hat{n})$ and n are then associated to the rolling motions (Σ) NS and (Σ) R respectively. This generalizes the rolling problems considered in Chitour and Kokkonen (Rolling manifolds and controllability: the 3D case, 2012) where both manifolds had the same dimension. The controllability issue is then addressed for both (Σ) NS and (Σ) R and completely solved for (Σ) NS . As regards to (Σ) R , basic properties for the reachable sets are provided as well as the complete study of the case $(n,\hat{n})=(3,2)$ and some sufficient conditions for non-controllability.

Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in (Mbele Bidima and Rásonyi in Ann. Oper. Res. 200:131–146, 2012), we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to 1 geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory. Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper.

The additive renormalization % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabs7adaWgaaWcbaGaaeySdiaab6cacaqG0bqefeKCPfgBaGqb% diaa-bcaaeqaaOGaeyypa0Jaa8hiaiaacIcacaaIYaGaeqiWdaNaai% ykamaaCaaaleqabaGaeyOeI0IaaGymaiaac+cacaaIYaaaaGqadOGa% a4hiaiGacwgacaGG4bGaaiiCaiaacIcacqGHsislcaqGXoWaaWbaaS% qabeaacaqGYaaaaOGaai4laiaaikdacaGGPaGaa4hiaiaacQdaciGG% LbGaaiiEaiaacchacqGHXcqSdaWadiqaaiabgkHiTiaadkeacaGGNa% GaaiikaiaadshacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaai4laiaa% ikdacaGFGaGaey4kaSIaa4hiaiaabg7acaWGcbGaai4jaiaacIcaca% WG0bGaaiykaaGaay5waiaaw2faaiaacQdaaaa!6C5C!\[{\rm{\delta }}_{{\rm{\alpha }}{\rm{.t}} } = (2\pi )^{ - 1/2} \exp ( - {\rm{\alpha }}^{\rm{2}} /2) :\exp \pm \left[ { - B'(t)^2 /2 + {\rm{\alpha }}B'(t)} \right]:\]is shown to be a generalized Brownian functional. Some of its properties are derived. is shown to be a generalized Brownian functional. Some of its properties are derived.

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with relation between the damping term and relaxation function. The second result is given without imposing any restrictive growth assumption on the damping term and the kernel function $g$ . Our result extends the decay result obtained for problems with finite history to those with infinite history.

This paper is concerned with the asymptotic analysis of space-velocity dependent thermostatted kinetic frameworks which include conservative, nonconservative and stochastic operators. The mathematical frameworks are integro-partial differential equations that can be proposed for the modeling of most phenomena occurring in biological and chemical systems. Specifically the paper focuses on the derivation of macroscopic equations obtained by performing a low-field and a high-field scaling into the thermostatted kinetic framework and considering the related convergence when the scaling parameter goes to zero. In the low-field limit, the macroscopic equations show diffusion with respect to both the space variable and a scalar variable that is introduced for the modeling of the strategy of the particle system. In the high-field limit, the macroscopic equations show hyperbolic behavior. The asymptotic analysis is also generalized to systems decomposed in various functional subsystems.