Connections between the Sznajd model with general confidence rules and graph theory

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabasi-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q > 2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the q-voter model).

Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations

In this paper, we give sufficient conditions for the uniform boundedness and uniform ultimate boundedness of solutions of a class of retarded functional differential equations with impulse effects acting on variable times. We employ the theory of generalized ordinary differential equations to obtain our results. As an example, we investigate the boundedness of the solution of a circulating fuel nuclear reactor model.

On the Hilbert transform in the framework of generalized functions

In this paper, a definition of the Hilbert transform operating on Colombeau's temperated generalized functions is given. Similar results to some theorems that hold in the classical theory, or in certain subspaces of Schwartz distributions, have been obtained in this framework.

Thermodynamics of Ising model with infinite-range interactions by generalized canonical ensemble

In this work we present the idea of how generalized ensembles can be used to simplify the operational study of non-additive physical systems. As alternative of the usual methods of direct integration or mean-field theory, we show how the solution of the Ising model with infinite-range interactions is obtained by using a generalized canonical ensemble. We describe how the thermodynamical properties of this model in the presence of an external magnetic field are founded by simple parametric equations. Without impairing the usual interpretation, we obtain an identical critical behaviour as observed in traditional approaches.

Encapsulation of small magnetic clusters in fullerene cages: A density functional theory investigation within van der Waals corrections

The encapsulation of magnetic transition-metal (TM) clusters inside carbon cages (fullerenes, nanotubes) has been of great interest due to the wide range of applications, which spread from medical sensors in magnetic resonance imaging to photonic crystals. Several theoretical studies have been reported; however, our atomistic understanding of the physical properties of encapsulated magnetic TM 3d clusters is far from satisfactory. In this work, we will report general trends, derived from density functional theory within the generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof (PBE), for the encapsulation properties of the TMm@C-n (TM = Fe, Co, Ni; m = 2-6, n = 60,70,80,90) systems. Furthermore, to understand the role of the van der Waals corrections to the physical properties, we employed the empirical Grimme's correction (PBE + D2). We found that both PBE and PBE + D2 functionals yield almost the same geometric parameters, magnetic and electronic properties, however, PBE + D2 strongly enhances the encapsulation energy. We found that the center of mass of the TMm clusters is displaced towards the inside C-n surfaces, except for large TMm clusters (m = 5 and 6). For few cases, e. g., Co-4 and Fe-4, the encapsulation changes the putative lowest-energy structure compared to the isolated TMm clusters. We identified few physical parameters that play an important role in the sign and magnitude of the encapsulation energy, namely, cluster size, fullerene equatorial diameter, shape, curvature of the inside C-n surface, number of TM atoms that bind directly to the inside C-n surface, and the van der Waals correction. The total magnetic moment of encapsulated TMm clusters decreases compared with the isolated TMm clusters, which is expected due to the hybridization of the d-p states, and strongly depends on the size and shape of the fullerene cages.

Ethanol and Water Adsorption on Close-Packed 3d, 4d, and 5d Transition-Metal Surfaces: A Density Functional Theory Investigation with van der Waals Correction

Nowadays, there is a great interest in the economic success of direct ethanol fuel cells; however, our atomistic understanding of the designing of stable and low-cost catalysts for the steam reforming of ethanol is still far from satisfactory, in particular due to the large number of undesirable intermediates. In this study, we will report a first-principles investigation of the adsorption properties of ethanol and water at low coverage on close-packed transition-metal (TM) surfaces, namely, Fe(110), Co(0001), Ni(111), Cu(111), Ru(0001), Rh(111), Pd(111), Ag(111), Os(0001), Ir(111), Pt(111), and Au(111), employing density functional theory (DFT) calculations. We employed the generalized gradient approximation with the formulation proposed by Perdew, Burke, and Erzenholf (PBE) to the exchange correlation functional and the empirical correction proposed by S. Grimme (DFT+D3) for the van der Waals correction. We found that both adsorbates binds preferentially near or on the on top sites of the TM surfaces through the 0 atoms. The PBE adsorption energies of ethanol and water decreases almost linearly with the increased occupation of the 4d and 5d d-band, while there is a deviation for the 3d systems. The van der Waals correction affects the linear behavior and increases the adsorption energy for both adsorbates, which is expected as the van der Waals energy due to the correlation effects is strongly underestimated by DFT-PBE for weak interacting systems. The geometric parameters for water/TM are not affected by the van der Waals correction, i.e., both DFT and DFT+D3 yield an almost parallel orientation for water on the TM surfaces; however, DFT+D3 changes drastically the ethanol orientation. For example, DFT yields an almost perpendicular orientation of the C-C bond to the TM surface, while the C-C bond is almost parallel to the surface using DFT +D3 for all systems, except for ethanol/Fe(110). Thus, the van der Waals correction decreases the distance of the C atoms to the TM surfaces, which might contribute to break the C-C bond. The work function decreases upon the adsorption of ethanol and water, and both follow the same trends, however, with different magnitude (larger for ethanol/TM) due to the weak binding of water to the surface. The electron density increases mainly in the region between the topmost layer and the adsorbates, which explains the reduction of the substrate work function.

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J (2) plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.

On Wald Residuals in Generalized Linear Models

A rigorous asymptotic theory for Wald residuals in generalized linear models is not yet available. The authors provide matrix formulae of order O(n(-1)), where n is the sample size, for the first two moments of these residuals. The formulae can be applied to many regression models widely used in practice. The authors suggest adjusted Wald residuals to these models with approximately zero mean and unit variance. The expressions were used to analyze a real dataset. Some simulation results indicate that the adjusted Wald residuals are better approximated by the standard normal distribution than the Wald residuals.

ON THE CONSISTENCY OF TWISTED GENERALIZED WEYL ALGEBRAS

A twisted generalized Weyl algebra A of degree n depends on a. base algebra R, n commuting automorphisms sigma(i) of R, n central elements t(i) of R and on some additional scalar parameters. In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for sigma(i) and t(i) are sufficient for the algebra to be nontrivial. However, in this paper we give all example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.

Adsorption of NO on the Rh-13, Pd-13, Ir-13, and Pt-13 Clusters: A Density Functional Theory Investigation

The adsorption of NO on transition-metal (TM) surfaces has been widely studied by experimental and theoretical techniques; however, our atomistic understanding of the interaction of nitrogen monoxide (NO) with small TM clusters is far from satisfactory, which compromises a deep understanding of real catalyst devices. In this study, we report a density functional theory study of the adsorption properties of NO on the TM13 (TM = Rh, Pd, Ir, Pt) clusters employing the projected augmented wave method. We found that the interaction of NO with TM13 is much more complex than that for NO/TM(111). In particular, for low symmetry TM13 clusters, there is a strong rearrangement of the electronic charge density upon NO adsorption and, as a consequence, the adsorption energy shows a very complex dependence even for adsorption sites with the same local effective coordination. We found a strong enhancement of the binding energy of NO to the TM13 clusters compared with the TM(111) surfaces, as the antibonding NO states are not occupied for NO/TM13, and the general relationship based on the d-band model between adsorption energy and the center of gravity of the occupied d-states does not hold for the studied TM13 clusters, in particular, for clusters with low symmetry. In contrast with the adsorption energy trends, the geometric NO/TM13 parameters and the vibrational N-O frequencies for different coordination sites follow the same trend as for the respective TM(111) surfaces, while the changes in the frequencies between different surfaces and TM13 clusters reflect the strong NO-TM13 interaction.

Platinum-Based Nanoalloys PtnTM55-n (TM=Co, Rh, Au): A Density Functional Theory Investigation

Structural and electronic properties of the PtnTM55-n (TM = Co, Rh, Au) nanoalloys are investigated using density functional theory within the generalized gradient approximation and employing the all-electron projected augmented wave method. For TM = Co and Rh, the excess energy, which measures the relative energy stability of the nanoalloys, is negative for all Pt compositions. We found that the excess energy has similar values for a wide range of Pt compositions, i.e., n = 20-42 and n = 28-42 for Co and Rh, respectively, with the core shell icosahedron-like configuration (n = 42) being slightly more stable for both Co and Rh systems because of the larger release of the strain energy due to the smaller atomic size of the Co and Rh atoms. For TM = Au, the excess energy is positive for all compositions, except for n = 13, which is energetically favorable due to the formation of the core-shell structure (Pt in the core and Au atoms at the surface). Thus, our calculations confirm that the formation of core-shell structures plays an important role to increase the stability of nanoalloys. The center of gravity of the occupied d-states changes almost linearly as a function of the Pt composition, and hence, based on the d-band model, the magnitude of the adsorption energy of an adsorbate can be tuned by changing the Pt composition. The magnetic moments of PtnCo55-n decrease almost linearly as a function of the Pt composition; however, the same does not hold for PtRh and PtAu. We found an enhancement of the magnetic moments of PtRh by a few times by increasing Pt composition, which we explain by the compression effects induced by the large size of the Pt atoms compared with the Rh atoms.