Extended crystal PDE’s stability,I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in some details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are carefully considered. (In the second part a stable extended crystal PDE, encoding anisotropic incompressible magnetohydrodynamics is obtained Ref. [A. Prástaro, Extended crystal PDE’s (submitted for publication)].)     

Geometry of PDE's. IV: Navier-Stokes equation and integral bordism groups

Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239-266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267-285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243-302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111-201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547-566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930-948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259-278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier-MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too.     

Extended crystal PDE’S stability,II: The extended crystal MHD-PDE’S

This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability, and its relationship with an important class of PDE’s called extended crystals PDE’s in the sense introduced in [A. Prástaro, Extended crystal PDE’s (submitted for publication)]. In fact, their integral bordism groups can be considered as extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I [A. Prástaro, Extended crystal PDE’s stability, I: The general theory, Math. Comput. Modelling, 49 (9–10) (2009) 1759–1780] we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions, finite time instabilities do not occur (stable extended crystal PDE’s). Here, we study in some detail, a new PDE encoding anisotropic incompressible magnetohydrodynamics. Stable extended crystal MHD-PDE’s are obtained, where in their smooth solutions, instabilities do not occur in finite time. These results are considered first for systems without a body energy source, and later, by also introducing a contribution from an energy source, in order to take into account nuclear energy production. A condition in order that solutions satisfy the second principle of thermodynamics is given.     

Electronic, chemical and structural characterization of CNTs grown by acetylene decomposition over MgO supported Fe-Co bimetallic catalysts

The electronic and chemical structure of carbon nanotubes synthesized by decomposition of acetylene over Fe-Co bimetallic catalysts in different growth conditions, were analyzed by valence band photoelectron spectroscopy and scanning electron microscopy. A clear relationship between the bonding features and the growth condition allowed us to determine the key parameters in terms of temperature, growth time and catalyst content. Furthermore, the analysis allowed a determination of the byproducts.     

Agent-based interpretations of classic network models

The paper is aimed at developing agent-based variants of traditional network models that make full use of concurrency. First, we review some classic models of the static structure of complex networks with the objective of developing agent-based models suited for simulating a large-scale, technology-enabled social network. Secondly, we outline the basic properties that characterize such networks. Then, we briefly discuss some classic network models and the properties of the networks they generate. Finally, we discuss how such models can be converted into agent-based models (i) to be executed more easily in heavily concurrent environments and (ii) to serve as basic blocks for more complex agent-based models. We evidence that many implicit assumptions made by traditional models regarding their execution environment are too expensive or outright impossible to maintain in concurrent environments. Consequently, we present the concurrency issues resulting from the violation of such assumptions. Then, we experimentally show that, under reasonable hypothesis, the agent-based variants maintain the main features of the classic models, notwithstanding the change of environment. Eventually, we present a meta-model that we singled out from the individual classic models and that we used to simplify the agent-oriented conversion of the traditional models. Finally, we discuss the software tools that we built to run the agent-based simulations.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

Molecular insights into azumamide e histone deacetylases inhibitory activity

Azumamide E, a cyclotetrapeptide isolated from the sponge Mycale izuensis, is the most powerful carboxylic acid containing natural histone deacetylase (HDAC) inhibitor known to date. In this paper, we describe design and synthesis of two stereochemical variants of the natural product. These compounds have allowed us to clarify the influence of side chain topology on the HDAC-inhibitory activity. The present contribution also reveals the identity of the recognition pattern between azumamides and the histone deacetylase-like protein (HDLP) model receptor and reports the azumamide E unprecedented isoform selectivity on histone deacetylases class subtypes. From the present studies, a plausible model for the interaction of azumamides with the receptor binding pocket is derived, providing a framework for the rational design of new cyclotetrapeptide-based HDAC inhibitors as antitumor agents.     

Island organization of TiO2 hierarchical nanostructures induced by surface wetting and drying

We report on the reorganization and bundling of titanium oxide nanostructured layers, induced by wetting with different solvents and subsequent drying. TiO(2) layers are deposited by pulsed laser deposition and are characterized by vertically oriented, columnar-like structures resulting from assembling of nanosized particles; capillary forces acting during evaporation induce bundling of these structures and lead to a micrometer-size patterning with statistically uniform islands separated by channels. The resulting surface is characterized by a hierarchical, multiscale morphology over the nanometer-micrometer length range. The structural features of the pattern, i.e., characteristic length, island size, and channel width, are shown to depend on properties of the liquid (i.e., surface tension) and thickness and density of the TiO(2) layers. The studied phenomenon permits the controlled production of multiscale hierarchically patterned surfaces of nanostructured TiO(2) with large porosity and large surface area, characterized by superhydrophilic wetting behavior without need for UV irradiation.