A nonlinear mathematical model to study the interactions of hot gases with cloud droplets and raindrops

In this paper, a nonlinear mathematical model is proposed and analyzed to study the interactions of hot gases with cloud droplets as well as with raindrops and their removal by rain from the stable atmosphere. The atmosphere, during rain, is assumed to consist of five nonlinearly interacting phases i.e. the vapour phase, the phase of cloud droplets, the phase of raindrops, the phase of hot gaseous pollutants and the absorbed phase of hot gases in the raindrops (if it exists). It is further assumed that these phases undergo ecological type growth and nonlinear interactions. The proposed model is analyzed using stability theory of differential equations and by numerical simulation. It is shown that the cumulative concentration of gaseous pollutants decreases due to rain and its equilibrium level depends upon the density of cloud droplets, the rate of formation of raindrops, emission rate of pollutants, the rate of falling absorbed phase on the ground, etc. It is noted here that if gases are very hot, cloud droplets are not formed and rain may not take place. In such a case gaseous pollutants may not be removed from the atmosphere due to non-occurrence of rain.     

LOCAL THEORY IN CRITICAL SPACES FOR COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE GASES

We are concerned with local existence and uniqueness of solutions for a general model of viscous and heat-conductive gases with low regularity assumptions on the initial data (the velocity and the temperature may be discontinuous). Local well-posedness is showed to hold in spaces which are critical with respect to the scaling of the equations, provided that the initial density is close enough to a positive constant. When initial data are a trifle more regular, local well-posedness holds for any initial density bounded away from zero. This former result lies on new estimates for linear heat equations with a non constant diffusion coefficient.     

MODELING THE DEPLETION OF A RENEWABLE RESOURCE BY POPULATION AND INDUSTRIALIZATION: EFFECT OF TECHNOLOGY ON ITS CONSERVATION

Abstract In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of a renewable resource by population and industrialization with resource‐dependent migration. The effect of technology on resource conservation is also considered. In the modeling process, four variables are considered, namely, density of a renewable resource, population density, density of industrialization, and technological effort. Both the growth rate and carrying capacity of resource biomass, which follows logistic model, are assumed to be simultaneously depleted by densities of population and industrialization but it is conserved by technological effort. It is further assumed that densities of population and industrialization increase due to increase in the density of renewable resource. The growth rate of technological effort is assumed to be proportional to the difference of carrying capacity of resource biomass and its current density. The model is analyzed by using the stability theory of differential equations and computer simulation. The model analysis shows that the biomass density decreases due to increase in densities of population and industrialization. It decreases further as the resource‐dependent industrial migration increases. But the resource may never become extinct due to population and industrialization, if technological effort is applied appropriately for its conservation.     

MODELING THE SPATIAL DISTRIBUTION OF THE ECONOMIC COSTS AND BENEFITS OF ILLEGAL GAME MEAT HUNTING IN THE SERENGETI

ABSTRACT. Illegal game meat hunting in the Serengeti National Park, Tanzania, and adjacent game reserves provides an important source of protein and cash income to local communities. We construct a profitability model that describes the spatial distribution of the economic costs and benefits of illegal hunting in the Serengeti during the late 1980s and early 1990s. Costs included capital investment in hunting weapons, W_R, and the opportunity cost of hunting, W_O, both held to be constants; and two spatially variable components, the logistic effort of traveling to hunting areas, W_L, and the penalties incurred if arrested, W_P. Benefit was the expected income from the sale of meat from resident wildlife species. The model suggests: (1) W_R is the most important cost. (2) W_L is the second most important cost and likely to determine the spatial distribution of hunting activity if hunters seekto minimize costs. (3) W_O and W_P are of minor importance, the former because alternative sources of income provide low pay, the latter because the overall chance of being arrested is low. (4) W_P exceeds W_L only in areas close to the boundary of protected areas. (5) Although resident wildlife contributes only a minor share of illegal offtake compared to the migratory herds, hunting resident wildlife is profitable in 68% of the area. This suggests that hunting of resident and migratory wildlife is highly profitable and may explain why the utilization of the target populations has become increasingly unsustainable.     

A kinetic model for polytropic gases with internal energy

We consider a mixture of N ideal, polytropic gases. Each species is described by a distribution function f_i(t, x, v, I) ≥ 0, 1 ≤ i ≤ N, defined on , and its evolution is governed by a Boltzmann-type equation. In order to recover the energy law of polytropic gases, the authors of [4] proposed a kinetic model in the framework of a weighted L¹ space. Another approach has been developed in [3] in the context of polyatomic gases. Following this previous lead, our model provides a L² framework in both variables v and I, to eventually perform a mathematical study of the diffusion asymptotics, as it was done in [2] for a model without energy exchange. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MODELING NUTRIENT (DISSOLVED INORGANIC NITROGEN) AND PLANKTON DYNAMICS AT SAGAR ISLAND OF HOOGHLY–MATLA ESTUARINE SYSTEM,WEST BENGAL,INDIA

Abstract Degradation of litter from mangrove forests adjacent to the creeks at Sagar Island of the Hooghly–Matla estuarine ecosystem is one of the principal sources of nutrient to the estuary. Nutrients augment the growth of phytoplankton, which in turn stimulates the production of zooplankton. Zooplankton serves as major food source for fish population of this estuarine system. Here, a dynamic model with three state variables (nutrient, phytoplankton, and zooplankton) is proposed using nitrogen (mgN/l) as currency. Input of dissolved inorganic nitrogen as nutrient, water temperature, surface solar irradiance, and salinity of upstream and downstream of the estuary, collected from the field, are incorporated as graph time functions in the model. Calibration and validation are performed by using collected data of two consecutive years. Model results indicate that the growth of zooplankton and phytoplankton are enhanced by increase in nutrient input in the system. Zooplankton biomass is affected by decrease in the salinity of the estuary. Sensitivity analysis results at ±10% indicate that maximum growth rate of phytoplankton (P_max) is the most sensitive parameter to the nutrient pool although growth rate of zooplankton (g_z) and half saturation constant for phytoplankton grazing by zooplankton (K_z) are most sensitive parameters to phytoplankton and zooplankton compartments, respectively. The model depicts the present status of plankton dynamics, which serve as major food resource for herbivorous and carnivorous fish species of the estuary. Effect of deforestation is tested in the model. Therefore, from management perspective, this model can be used to predict the impact of mangroves on nutrient and plankton dynamics, which will give complete information of both shell and fin fish productions in the estuary.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

Weighted Polynomial Approximation for Weights with Slowly Varying Extremal Density

Polynomial approximation by weighted polynomials of the form wⁿ(x) P_n(x) is investigated on closed subsets of the real line. It is known that the possibility of approximation is closely related to the density of an extremal measure associated with w via a weighted energy problem. It is also known that if in a neighborhood of a point x₀ this density is continuous and positive, then, in that neighborhood, any continuous function can be approximated. The aim of the present paper is twofold. On the one hand it is shown that the same approximation theorem is true if in a neighborhood of x₀ the density is slowly varying and is bounded away from 0. This allows singularities of logarithmic types. On the other hand, we also show that under some mild conditions, if the density at x₀ is slowly varying, then approximation is still possible even if the density vanishes at x₀ . This is the first positive result for approximation with a vanishing density.     

Thermal properties of bodies in fractal and cantorian physics

Fundamental laws describing the heat diffusion in fractal environment are discussed. It is shown that for the three-dimensional space the heat radiation process occur in structures with fractal dimension D 0,1), whereas in structures with D (1,3 heat conduction and convection have the upper hand (generally in the real gases).To describe the heat diffusion a new law has been formulated. Its validity is more general than the Plank’s radiation law based on the quantum heat diffusion theory. The energy density w = f (K, D), where K is the fractal measure and D is the fractal dimension exhibit typical dependency peaking with agreement with Planck’s radiation law and with the experimental data for the absolutely black body in the energy interval kT < Kc. The positions of the energy density maximums (for fractal dimensions D_m < 0.31854) are in a good agreement with the maximums determined by Wien’s displacement law with the help of the Lambert’s W- Function u(A) = A + W[−Aexp(−A)], where A ≈ 1.9510 and u = hc/λ_mkT_m ≈ 1.5275. The agreement of the fractal model with the experimental outcomes is documented for the spectral characteristics of the Sun. The properties of stellar objects (black holes, relict radiation, etc.) and the elementary particles fields and interactions between them (quarks, leptons, mesons, baryons, bosons and their coupling constants) will be discussed with the help of the described mathematic apparatus in our further contributions.The general gas law for real gases in its more applicable form than the widely used laws (e.g. van der Waals, Berthelot, Kammerlingh–Onnes) has been also formulated. The energy density, which is in this case represented by the gas pressure p = f (K, D), can gain generally complex value and represents the behaviour of real (cohesive) gas in interval D (1,3. The gas behaves as the ideal one only for particular values of the fractal dimensions (the energy density is real-valued). Again, it is shown that above the critical temperature (kT > Kc) and for fractal dimension D_m > 2.0269 the results are comparable to the kinetics theory of real (ideal) gas (van der Waals equation of state, compressibility factor, Boyle’s temperature). For the critical temperature (Kc = kT_r) the compressibility factor gains Z = 1 (except for the ideal gas case D = 3) also for the fractal dimension D = 1/ = 1.618033989, where is the golden mean value of the El Naschie’s golden mean field theory. To determine the minimum it is also possible to employ the Lambert’s W− Function u(A) = A + W[−Aexp(−A)], whereA ≈ 0.6779 and u ≈ −0.7330. The thermal properties of fractal structures (thermal capacity, thermal conductivity, diffusivity) and additional parameters (enthalpy, entropy, etc.) will be defined using the mathematic apparatus in the future. Good agreement of the fractal model with experimental data is documented on the compressibility factor of various gases.     

Optimal production mix

This paper deals with a production plant in which two different products can be produced. The plant consists of three subsystemsS _i . Before or after a phase of separate processing in subsystemsS ₁ andS ₂, the two products have to be processed in subsystemS ₃. Each of these subsystems has a limited capacity.In the first part, we assume empty stocks at the beginning; at a fixed timeT in the future, certain quantitiesX _i of the two products have to be delivered to the customers. Facing linear holding costs, convex production costs, and stringent capacity constraints, the problem is to decide when to produce which product at what rate.It is shown that the optimal solution consists of up to six different regimes and that the time paths of the production rates need not be monotonic. These results, which can be obtained analytically, are also illustrated in several numerical examples.Finally, the case is considered where the terminal demand at timeT is replaced by a continuous and seasonally fluctuating demand rate. It is demonstrated that the optimal production rates show an interesting and nontrivial behavior. In particular, it may happen that, on intervals where the demand for the one product increases, the optimal production rate decreases. This is also demonstrated by computer plots in some numerical examples.The first author gratefully acknowledges support from the Austrian Science Foundation under Grant S3204 and the second author from Stiftung Volkswagenwerk. An earlier version of this paper was presented at the DGOR-NSOR Joint Conference, Eindhoven, Holland, September 23–25, 1987.     

UOBYQA: unconstrained optimization by quadratic approximation

UOBYQA is a new algorithm for general unconstrained optimization calculations, that takes account of the curvature of the objective function, F say, by forming quadratic models by interpolation. Therefore, because no first derivatives are required, each model is defined by ?(n+1)(n+2) values of F, where n is the number of variables, and the interpolation points must have the property that no nonzero quadratic polynomial vanishes at all of them. A typical iteration of the algorithm generates a new vector of variables, _t say, either by minimizing the quadratic model subject to a trust region bound, or by a procedure that should improve the accuracy of the model. Then usually F( _t ) is obtained, and one of the interpolation points is replaced by _t . Therefore the paper addresses the initial positions of the interpolation points, the adjustment of trust region radii, the calculation of _t in the two cases that have been mentioned, and the selection of the point to be replaced. Further, UOBYQA works with the Lagrange functions of the interpolation equations explicitly, so their coefficients are updated when an interpolation point is moved. The Lagrange functions assist the procedure that improves the model, and also they provide an estimate of the error of the quadratic approximation to F, which allows the algorithm to achieve a fast rate of convergence. These features are discussed and a summary of the algorithm is given. Finally, a Fortran implementation of UOBYQA is applied to several choices of F, in order to investigate accuracy, robustness in the presence of rounding errors, the effects of first derivative discontinuities, and the amount of work. The numerical results are very promising for n≤20, but larger values are problematical, because the routine work of an iteration is of fourth order in the number of variables. Received: December 7, 2000 / Accepted: August 31, 2001?Published online April 12, 2002     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

BestL 1 approximation by convex functions

It is shown that a function inL ₁ has a best approximation by convex functions, and that the net of bestL _p approximations converges asp decreases to one.     

MODELING EFFECTS OF PRIMARY AND SECONDARY TOXICANTS ON RENEWABLE RESOURCES

ABSTRACT. In this paper a nonlinear mathematical model to study effects of primary and secondary toxicants on the biomass of resources such as forestry, agricultural crops, etc., is proposed and analyzed. The primary toxicant is emitted into the environment with a constant prescribed rate by an external source and a part of which is transformed into a secondary toxicant, which is more toxic, both affecting the resource simultaneously. By using stability theory of differential equations, it is shown that the biomass density of resource attains an equilibrium level, the magnitude of which is smaller than its original (toxicant independent) carrying capacity and it decreases as the emission rate of primary toxicant increases. It is also shown that the decrease in biomass density of resource is more than the corresponding case of a single toxicant due to large transformation and uptake rates and high toxicity of secondary toxicant. It is pointed out that the resource may even become extinct if emission rate of primary toxicant and transformation rate of secondary toxicant are very large and their effects on resource are sufficiently harmful due to large uptake and high toxicity of secondary toxicant which is more toxic.     

On universal Banach spaces of density continuum

We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density at most continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ_∞/c ₀ is such a space if we assume the continuum hypothesis. Some additional set-theoretic assumption is indeed needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.     

MODELING THE POPULATION DYNAMICS OF A SUBTROPICAL UNGULATE IN A VARIABLE ENVIRONMENT: RAIN,COLD AND PREDATORS

ABSTRACT. A structured population model was developed for a large ungulate, the kudu (Tragelaphus strepsiceros). From a ten-year study in South Africa's Kruger National Park, relationships were established between annual survival rates of particular age classes and resource availability indexed by the ratio between annual rainfall and population biomass density. The projected population dynamics resembled that from a simple logistic model, but with the convexity of density dependence and intrinsic growth rate dependent upon assumptions about how age-specific mortality changed at low density levels. Moreover, rather than being a preset constant, the effective carrying capacity K wasa dynamic variable dependent upon rainfall. The model closely replicated the observed dynamicsof the kudu population over the study period, but failed to predict the observed kudu density at the start of the study from prior rainfall alone. Episodic cold weather extremeswere identified ashaving an additional influence on kudu dynamics. The model was also unsuccessful in predicting the changesin kudu abundance that occurred in Kruger Park subsequent to the study. Here changes in predation perhaps due to predator switching were a possible influence. These additional factorsinfluencing population dynamicswould not have been recognized without first establishing the effects of changing resource availability in response to rainfall fluctu-ationsbetween years. The elaborated model incorporating the effects of resource supply as influenced by rainfall, density dependence, background predation pressure and episodic severe weather hasbroader reliability than simpler modelsfor conservation applications, while still having a firm empirical foundation.     

OPEN ACCESS AND EXTINCTION OF THE PASSENGER PIGEON IN NORTH AMERICA

ABSTRACT. An open access model is formulated where X is a renewable resource and E is the level of effort devoted to harvest. Net growth is assumed to exhibit critical depensation and the open access system is described by two nonlinear differential equations where r > 0 is the intrinsic growth rate, K1 is the minimum viable population level, K₂ is the environmental carrying capacity, K₂ > K₁ > 0, q > 0 is the catchability coefficient, ? > 0 is an adjustment coefficient, (p – s) > 0 is the market price net of shipping cost, and c > 0 is the unit cost of effort at the harvest site. It is shown that the E= 0 isocline is a vertical line at X∞=c/[(p‐ s)q] and that the open access system passes through a supercritical Hopf bifurcation as X∞ moves from a level above (K₁+K₂)/2 to a level below (K₁+K₂)/2. For X∞ above (K₁+K₂)/2 the open access equilibrium is locally stable. For X∞ below (K₁+K₂)/2 the open access equilibrium will be locally unstable. At X∞=(K₁+K₂)/2 the system has a stable limit cycle. This analysis is useful in interpreting the economic history of the passenger pigeon. The limited empirical evidence would suggest that X∞=c/[(p – s)q] declined below (K₁+K₂)/2 during the last half of the 19th century as a result of improved rail transport and communications (the telegraph). It is thought that the passenger pigeon was extinct in the wild by 1901. The last passenger pigeon died in captivity at the Cincinnati Zoological Gardens on September 1, 1914. X=rX(X/ K₁– 1)(1 –X/K₂)–qXE and E=α[(p – s)qXE – cE],     

Growth of Cu2Cl2 and Cu2Br2 on single crystal faces and polycrystalline copper during dissolution in CuSO4 + H2SO4

Cu₂Cl₂ and Cu₂Br₂ precipitate and crystallise in the form of triangular pyramids and dendrites when Cu is immersed in aerated, acid CuSO₄ solution containing HC1 or HBr. The critical concentration of HC1 or HBr for this precipitation depends on the surface of the copper: Poly-crystalline < (110) < (100) < (111). In the deaerated solution there is no precipitation of Cu₂Cl or Cu₂Br₂ even at a high concentration of HC1 or HBr when only preferential etching occurs.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

The impact of constant effort harvesting on the dynamics of a discrete‐time contest competition model

In this paper, we study a general discrete‐time model representing the dynamics of a contest competition species with constant effort exploitation. In particular, we consider the difference equation x _{n +1}=x _n f (x _{n ?k} )?h x _n where h >0, k ∈{0,1}, and the density dependent function f satisfies certain conditions that are typical of a contest competition. The harvesting parameter h is considered as the main parameter, and its effect on the general dynamics of the model is investigated. In the absence of delay in the recruitment (k =0), we show the effect of h on the stability, the maximum sustainable yield, the persistence of solutions, and how the intraspecific competition change from contest to scramble competition. When the delay in recruitment is 1 (k =1), we show that a Neimark‐Sacker bifurcation occurs, and the obtained invariant curve is supercritical. Furthermore, we give a characterization of the persistent set.