Infra-red spectral studies of CHF2-CHCl2 and CBrCl2-CBrCl2

The infra-red spectra of two polyhalogenated ethanes, 1, 1-difluoro-2, 2-dichloro ethane and 1, 2-dibromo-1, 1, 2, 2-tetrachloro ethane were obtained in the region of 2·5–38 microns. The spectra of CHF₂-CHCl₂ were obtained in gas, liquid and solid states and also in solvents of widely varying dielectric constants. These spectral studies showed that the molecule exists in the form of two rotational isomers. The fundamental modes of the two isomers were identified on the basis of intensity changes from gas to liquid to solid and in solvents of different dielectric constants. The observed infra-red absorption bands have been assigned in terms of the fundamental absorption bands. Spectra of CBrCl₂-CBrCl₂ were obtained in the solid and vapour phase and in various solvents. These spectra showed that CBrCl₂-CBrCl₂ exists only in the trans form. The infra-red spectra, combined with Raman data available from literature was used for complete assignment of the observed absorption bands.     

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2K₂-partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2K₂-partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2K₂-hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.     

Radiation balance of the lower stratosphere

A study of the radiation balance of the atmosphere involves the determination of the absorption and emission of radiant energy due to the different constituents of the atmosphere. In the stratosphere below 50 km., these are mainly ozone, CO₂ and water vapour. The major part of the absorbed energy is from solar radiation, the actual absorption at different levels being determined by the absorption coefficients of these gases and their vertical distributions. In this paper, a detailed survey is made of all the available data and of the recent methods developed for using such data, and after proper selection, curves are prepared giving the solar energy absorbed by different quantities of O₃, CO₂ and H₂O. These are used to calculate the absorption of solar energy per unit volume and per unit mass in different 2 km. layers for certain vertical distributions of the constituents. It is seen that the mass density of absorption due to ozone above 50 km. becomes much larger than that due to CO₂ and H₂O while in the region below 30 km., they become comparable. Water vapour becomes more and more effective as we approach the earth.     

Baire functions and spaces of Baire functions

The paper considers (1) the tightness of spaces of Baire functions and their subspaces endowed with the topology of pointwise convergence; (2) Z _σ-mappings of K-analytic spaces; (3) K _σ-analytic spaces (Tychonoff spaces that are Z _σ-images of K-analytic spaces). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 4, pp. 3–39, 2003.     

Cyclic-order graphs and Zarankiewicz's crossing-number conjecture

Zarankiewicz's conjecture, that the crossing number of the complete-bipartite graph K_{m, n} is [1/2 m] [1/2 (m] ?1) [1/2 n] [1/2 (n ?1)], was proved by Kleitman when min(m, n) ≤ 6, but was unsettled in all other cases. The cyclic-order graph COn arises naturally in the study of this conjecture; it is a vertex-transitive harmonic diametrical (even) graph. In this paper the properties of cyclic-order graphs are investigated and used as the basis for computer programs that have verified Zarankiewicz's conjecture for K_7,7 and K_7,9; thus the smallest unsettled cases are now K_7,11 and K_9,9. © 1993 John Wiley & Sons, Inc.     

Carbon-dioxide storage and phase transitions: towards an understanding of crack development in the cap-rock layer

Supercritical CO₂ can be injected into deep saline aquifers to reduce the amount of CO₂ in the atmosphere and thus, lessen the impact on the global warming. Qualified reservoirs should be in a sufficient depth to guarantee the thermodynamical environment for the supercritical state of CO₂ and should be confined by an impermeable cap-rock layer. It is crucial to guarantee the safety of the storage site. Therefore, deformation processes and crack development of the rock matrix and the cap-rock layer, which might be induced by the high pressure injection of CO₂, must be investigated. If cracks occur, CO₂ could migrate into shallower regions, where the temperature and pressure cannot support the supercritical condition of the CO₂ anymore. Thus, it is important to describe the phase transition process between supercritical, liquid and gaseous CO₂. The Theory of Porous Media (TPM), see e. g. [1], provides a useful continuum-mechanical basis to describe real natural systems in a thermodynamically consistent way. Hence, the TPM is applied to model multiphasic flow of CO₂ and water and to include elasto-plastic solid deformations of the porous matrix. The Peng-Robinson equation, e. g. [2], is implemented as a cubic equation of state to describe the phase behaviour of CO₂. However, the two-phase region cannot be represented by a continuously differentiable function such as the Peng-Robinson equation and thus, the Antoine equation provides additional information of the vapourisation curve. The Extended Finite-Element Method (XFEM) will be used to account for the discontinuities arising from crack development due to solid deformations [3]. Herein, special attention has to be paid to the matrix-fracture interaction of the fluid phases. Numerical examples are performed to investigate the injection of CO₂ into a saline aquifer. These are computed with the FE program PANDAS, which allows for solutions of strongly coupled multiphasic problems in deformable porous media. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The emissions trading paradox

This article considers the price history of CO₂ allowances in the EU Emission Trading Scheme. Since European Emissions Trading started in 2005, the prices of allowances have varied between less than one and thirty Euro per ton of CO₂. This previously unpredicted volatility and, more notably, a significant price crash in May 2005 led to the hypothesis that electricity producers might use their market power to influence the prices of allowances. Besides market power, the combination of information asymmetry and price interdependencies (between prices of primary goods – especially electricity – and allowances) plays an important role in explaining the emissions trading paradox. The model presented will show that banking can lead to such a price crash if market participators act rationally. Furthermore, in such a scenario banking can be profitable for sellers at the cost of buyers.     

On the prescribing �� 2 curvature equation on {\mathbb S^4}

Prescribing ?? _k curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. For a given a positive function K to be prescribed on the 4-dimensional round sphere, we obtain asymptotic profile analysis for potentially blowing up solutions to the ?? ₂ curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. Under the same non-degeneracy condition on K, we also prove uniform a priori estimates for solutions to a family of ?? ₂ curvature equations deforming K to a positive constant; and under an additional, natural degree condition on a finite dimensional map associated with K, we prove the existence of a solution to the ?? ₂ curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.     

Generalized gluing for Einstein constraint equations

In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M ₁, g ₁, Π₁) and (M ₂, g ₂,Π₂) along a common isometrically embedded k-dimensional sub-manifold (K, g _K ). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction produces a polyneck between M ₁ and M ₂ whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : =  m − k of K in M ₁ and M ₂ is required to be  ≥  3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat.     

Existence of positive solutions for a second-order ordinary differential system

In this paper we consider the existence of positive solutions for a second-order ordinary differential system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing a cone K₁×K₂ which is the Cartesian product of two cones in space C[0,1] and computing the fixed point index in K₁×K₂, we establish the existence of positive solutions for the system. We remark that, differently from the literature, we deal with our problem on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

Synchronization dynamics in a ring of four mutually coupled biological systems

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourth-order Runge–Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K₁, K₂) plane.     

G-Strands

A G-strand is a map g(t,s):?×?→G for a Lie group G that follows from Hamilton’s principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3) _K -strand dynamics for ellipsoidal rotations is derived as an Euler–Poincaré system for a certain class of variations and recast as a Lie–Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) _K -strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the Sp(2)-strand. The Sp(2)-strand is the G-strand version of the Sp(2) Bloch–Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. Diff(?)-strand equations on the diffeomorphism group G=Diff(?) are also introduced and shown to admit solutions with singular support (e.g., peakons).     

Simulation of a reacting gas–liquid bubbly flow with CFD and PBM: Validation with experiments

In this work we use computational fluid dynamics (CFD) to simulate a reactive gas–liquid bubbly system in a rectangular bubble column, operating at low superficial velocities (i.e. homogeneous regime). The gas bubbles, injected in the column through a sparger, contain one of the reactants, namely CO₂, that via mass transfer moves to the continuous liquid phase, where it reacts with NaOH. A key role is played by the bubble size distribution (BSD) and the specific surface area that define the overall mass transfer rate in the CFD model. In order to correctly predict the BSD and the polydispersity of the bubbly system the population balance equation is solved by the quadrature method of moments (QMOM), within the OpenFOAM (v. 2.2.x) two-fluid solver compressibleTwoPhaseEulerFoam. To reduce the computational time and increase stability, a second-order operator-splitting technique for the solution of the chemically reactive species is also implemented, allowing to solve the different processes involved with their own time-scale. To our knowledge this is the first time that QMOM is employed for the simulation of a real reactive bubbly system and predictions are validated against experiments.     

Analysis of the Adhesion and Adhesion Strength of Two-Layer Polymer Films and Coatings

The formation of adhesion bonds during the creation of two-layer polymer films and coatings capable of acid–base interaction is considered. The coatings are obtained from polymer solutions, and the second layer is formed from the polymer solutions and their mixtures on a hard polymer coating. As a result of absorption by the coating of a considerable amount of solvents (to 14%), polymer macrochains are able to approach one another at a distance necessary for the formation of acid–base (hydrogen) bonds and a transition layer. The polar component of the work of adhesion is equal to about 50% of the whole work W _a. There is no interrelation of the thermodynamic work W _a between the films and the work of their separation into layers. The main part of the external work upon the separation is spent (if strong donor–acceptor bonds are formed) for extracting the segments and macrochains of polymers out of the films.     

Existence of three nontrivial solutions for an elliptic system

In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in K₁, K₂ and K₁×K₂, we obtain that the elliptic system has three nontrivial solutions (u,0), (0,v) and (u^∗,v^∗). It is remarkable that the third nontrivial solution (u^∗,v^∗) is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

On DRC‐covering of Kn by cycles

This paper considers the cycle covering of complete graphs motivated by the design of survivable WDM networks, where the requests are routed on INF‐networks which are protected independently from each other. The problem can be stated as follows: for a given graph G, find a cycle covering of the edge set of K_n, where V(K_n) = V(G), such that each cycle in the covering satisfies the disjoint routing constraint (DRC7rpar;, relatively to G, which can be stated as follows: to each edge of K_n we associate in G a path and all the paths associated to the edges of a cycle of the covering must be vertex disjoint. Here we consider the case where G = C_n, a ring of size n and we want to minimize the number of cycles in the covering. We give optimal solutions for the problem as well as for variations of the problem, namely, its directed version and the case when the cycle length is fixed to 4. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 100–112, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10040     

Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations

This paper is devoted to the study of multiple and single positive solutions of two-point boundary value problems for nonlinear second-order singular and impulsive differential systems. By constructing a cone _K1×_K2$K_1\times K_2$, which is the Cartesian product of two cones in the space C[0,1]$C[0,1]$, and computing the fixed-point index in the _K1×_K2$K_1\times K_2$, we establish the conditions for the existence of at least one or at least two positive solutions to two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations.