A note on the Morse homology for a class of functionals in Banach spaces involving the pp-Laplacian

In this paper we show how to construct Morse homology for an explicit class of functionals involving the $p$-Laplacian. The natural domain of definition of such functionals is the Banach space $W^{1,2p}_0(\Omega)$, where $p>n/2$ and $\Omega \subset \mathbb R^n$ is a bounded domain with sufficiently smooth boundary. As $W^{1,2p}_0(\Omega)$ is not isomorphic to its dual space, critical points of such functionals cannot be non-degenerate in the usual sense, and hence in the construction of Morse homology we only require that the second differential at each critical point be injective. Our result upgrades a result of Cingolani and Vannella, where critical groups for an analogous class of functionals are computed, and provides in this special case a positive answer to Smale's suggestion that injectivity of the second differential should be enough for Morse theory

The Index Bundle and Multiparameter Bifurcation for Discrete Dynamical Systems

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author

Specificity of logistics processes in industrial waste management. Part 1, Organizational approach

Problematyka gospodarowania odpadami, a w szczególności odpadami przemysłowymi, stanowi istotne wyzwanie dla współczesnej gospodarki. Mając na uwadze uwzględnianie priorytetów zrównoważonego rozwoju organizacja procesów związanych z zarządzaniem przepływami odpadów przemysłowych musi uwzględniać w głównej mierze procesy logistyczne. W tym kontekście stanowią one nieodłączny element poprawnego funkcjonowania gospodarki odpadami. Artykuł prezentuje modelowe podejście do interpretacji procesów logistycznych związanych z poprawną realizacją gospodarki odpadami przemysłowymi.The issue of waste management, and in particular industrial waste, represents a significant challenge for today's economy. Given the priorities of sustainable development taking into account the organization of the processes related to the management of industrial waste flows must take into account mainly logistics processes. In this context, they are an integral part of the proper functioning of waste management. This paper presents a model approach to the interpretation of logistics processes associated with the correct implementation of the industrial waste management

Specificity of logistics processes in industrial waste management. Part 2, Costs approach

Gospodarowanie odpadami przemysłowymi, realizowane w sposób poprawny i zgodny z regulacjami prawnymi w tym zakresie, uwzględnia procesy logistyczne związane z przepływami tych odpadów. Oprócz aspektów organizacyjnych w tym zakresie, równie istotne znaczenie mają koszty dotyczące tych procesów. Artykuł prezentuje model matematyczny uwzględniający koszty procesów logistycznych w gospodarowaniu odpadami przemysłowymi przez wyspecjalizowane w tego rodzaju usługach przedsiębiorstwach.Industrial waste management, carried out in a correct and compliant with legal regulations in this area includes logistics processes associated with the waste flow. In addition to the organizational aspects in this regard, just as important are the costs of these processes. This paper presents a mathematical model taking into account the costs of logistics processes in the industrial waste management by specializing in this kind of services companies

Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

In this paper we introduce a new compactness condition Property - (C) - for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property -(C) flows and a Conley index can be defined. An important class of flows satisfying the this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse-Conley-Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index. (c) 2017 Elsevier Inc. All rights reserved