Infinite Values Research Articles

Quasi-Monte Carlo methods with applications in finance

We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

Activity coefficients at infinite dilution measurements for organic solutes and water in the ionic liquid 4-methyl- N-butyl-pyridinium bis(trifluoromethylsulfonyl)-imide

The activity coefficients at infinite dilution, γ 13 ∞ for 36 solutes: alkanes, cycloalkanes, alkenes, alkynes, aromatic hydrocarbons, alcohols, thiophene, tetrahydrofuran, ethers, acetone, and water in the ionic liquid 4-methyl- N-butyl-pyridinium bis(trifluoromethylsulfonyl)-imide [bmPY][NTf 2] were determined by gas–liquid chromatography at temperatures from 298.15 K to 368.15 K. The partial molar excess enthalpies at infinite dilution values Δ H 1 E , ∞ were calculated from the experimental γ 13 ∞ values obtained over the temperature range. The selectivity for different separation problems were calculated from the γ 13 ∞ and compared to the literature values for other ionic liquids, N-methyl-2-pyrrolidinone (NMP) and sulfolane.

New inductive constructions of complete caps inPG(N, q),qeven

AbstractSome new families of small complete caps inPG(N, q),qeven, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap inPG(N, q),N≥4, for allq≥23. In particular, substantial improvements are obtained for infinite values ofqsquare, includingq=22Cm,C≥5,m≥3; forq=2Cm,C≥5,m≥9, withC,modd; and for allq≤218. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010

Apparent molar volumes and isentropic compressibilities of benzyltrialkylammonium chlorides in water at (293.15, 303.15, 313.15, 323.15, and 333.15) K

Densities and ultrasound speeds of benzyltrialkylammonium chlorides (BTAACls) were measured accurately in aqueous solutions at five temperatures (293.15, 303.15, 313.15, 323.15, and 333.15) K. The data were utilized in determining apparent molar volumes, V Φ , and apparent molar isentropic compressibilities, K SΦ . Infinite dilution values of these apparent molar quantities, V Φ ∘ and K S Φ ∘ , were determined by extrapolation procedures. Contribution of CH 2 groups, along the alkyl groups of BTAACls, to V Φ ∘ and K S Φ ∘ values were derived from plots of these quantities as a function of number of additional CH 2 groups in going from benzyltrimethylammonium chloride to benzyltributylammonium chloride. The temperature dependencies of V Φ ∘ and K S Φ ∘ values were examined on the basis of isobaric expansivity. Apparent molar isobaric expansivities at infinite dilution, E Φ ∘ , were obtained from the slopes of V Φ ∘ versus temperature data. Concentration dependencies of V Φ and K SΦ were examined. A fair correlation was observed between V Φ ∘ and K S Φ ∘ values of the three BTAACls.

Density, sound velocity and viscosity properties of aqueous sodium metatungstate solutions and an application of these solutions in heavy mineral separations

Densities, sound velocities and viscosities of aqueous sodium metatungstate (SMT) solutions, medium for heavy mineral separations, have been measured accurately in a concentration range from 0.0055 mol kg − 1 to 1.73 mol kg − 1 at 288.15, 293.15, 298.15, 303.15 and 308.15 K at 1 atm. The data were used in evaluating thermodynamic properties as apparent molar volumes, apparent molar isentropic compressibilities and apparent molar expansivities. Infinite dilution values of these properties were determined using appropriate extrapolation procedures utilizing Debye-Hückel limiting law for electrolyte solutions. Apparent molar volumes were found to decrease with concentration and increase with temperature. The data of apparent molar volumes at infinite dilution as a function of temperature were found to fit excellently to a quadratic equation. Apparent molar expansivities at various temperatures were determined from the derivative of this equation. Apparent molar isentropic compressibilities were found to be negative and to increase with temperature. The results were interpreted in term of solute–solvent interactions. Heavy mineral separation process using SMT at a density of 2.89 g cm − 3 was applied on fine grained sand samples taken from Goynuk region of southwest Turkey.

RISK MEASURES FOR NON‐INTEGRABLE RANDOM VARIABLES

We show that when a real‐valued risk measure is defined on a solid, rearrangement invariant space of random variables, then necessarily it satisfies a weak compactness, also called continuity from below, property, and the space necessarily consists of integrable random variables. As a result we see that a risk measure defined for, say, Cauchy‐distributed random variable, must take infinite values for some of the random variables.

Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind of a computer–the Infinity Computer–able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given.

Activity coefficients at infinite dilution measurements for organic solutes and water in the ionic liquid triethylsulphonium bis(trifluoromethylsulfonyl)imide

The activity coefficients at infinite dilution, γ 13 ∞ for 32 solutes: alkanes, alkenes, alkynes, cycloalkanes, aromatic hydrocarbons, alcohols, thiophene, tetrahydrofuran, tert-butyl methyl ether, and water in the ionic liquid triethylsulphonium bis(trifluoromethylsulfonyl)imide [Et 3S][NTf 2] were determined by gas–liquid chromatography at the temperatures from T = 298.15 K to T = 368.15 K. The partial molar excess enthalpies at infinite dilution values Δ H 1 E, ∞ were calculated from the experimental γ 13 ∞ values obtained over the temperature range. The selectivities for the n-hexane/benzene and cyclohexane/benzene, n-hexane/thiophene, cyclohexane/tiophene, benzene/tiophene, and hexen-1-ene/ n-hexane separation problems were calculated from the γ 13 ∞ and compared to the literature values for other ionic liquids, NMP, and sulfolane.

Pressure gradients, shell-crossing singularities and acoustic oscillations – application to inhomogeneous cosmological models

Abstract Inhomogeneous cosmological models have recently become a very interesting alternative to standard cosmology. This is because these models are able to fit cosmological observations without the need for dark energy. However, due to inhomogeneity and pressureless matter content, these models can suffer from shell-crossing singularities. These singularities occur when two shells of dust collide with each other leading to infinite values of the density. In this Letter, we show that if inhomogeneous pressure is included then these singularities can be prevented from occurring over the period of structure formation. Thus, a simple incorporation of a gradient of pressure allows for more comprehensive studies of inhomogeneous cosmological models and their application to cosmology.

Magnetic phase diagrams of manganite-like local-moment systems with Jahn-Teller distortions

We use an extended two-band Kondo lattice model (KLM) to investigate the occurrence of different (anti)ferromagnetic phases or phase separation depending on several model parameters. With regard to CMR-materials such as the manganites we have added a Jahn-Teller term, direct antiferromagnetic coupling, and Coulomb interaction to the KLM. The electronic properties are self-consistently calculated in an interpolating self-energy approach with no restriction to classical spins and going beyond mean-field treatments. Further on we do not have to limit the Hund's coupling to low or infinite values. Zero-temperature phase diagrams are presented for large parameter intervals. There are strong influences of the type of Coulomb interaction (intraband, interband) and of the important parameters (Hund's coupling, direct antiferromagnetic exchange, Jahn-Teller distortion), especially at intermediate couplings.

An approximate Riemann solver for sensitivity equations with discontinuous solutions

The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach.

One-Dimensional Time-Dependent Frustrated and Disordered Ising Model

We study the combined effects of disorder and frustration on time dependences of thermodynamical quantities. We consider the kinetic Ising model in the version proposed by Glauber [1]. This model has already been considered at different levels of complexity. Admission of a possibility of the exchange integrals taking random values complicates the problem, although it does not introduce competition, but it can still be analytically investigated ([2] and [3]). Additional complexity appears when frustration is admitted. The simplest model with frustration is the one-dimensional antiferromagnetic Ising model in a magnetic field. However, in this model only for zero magnetic field it is possible to derive expressions for the time dependences of thermodynamic quantities, as only then the time derivative of the n-spin correlation function can be expressed in terms of the corresponding n-spin correlation functions. For other models the equation of motion of the n-spin correlation function includes also other correlations, which means that all 2 correlation functions must be taken into regard to derive an expression for anyone of them. In literature this rule is known as the BBGKY hierarchy [4]. In certain particular cases the disordered one-dimensional Ising model has been analysed in an external magnetic field. The case when the magnetic field H can be assumed to tend to zero has been studied by Jose et al. [5]. A radically different situation of the random fields of infinite values when the system

Activity Coefficients at Infinite Dilution Measurements for Organic Solutes and Water in the Ionic Liquid 1-Butyl-3-methylimidazolium Trifluoromethanesulfonate

The activity coefficients at infinite dilution, gamma 13 (infinity) for 32 solutes: alkanes, alkenes, alkynes, cycloalkanes, aromatic hydrocarbons, alcohols, thiophene, tetrahydrofurane, tert-butyl methyl ether, and water in the ionic liquid 1-butyl-3-methylimidazolium trifluoromethanesulfonate [BMIM][CF3SO3] were determined by gas-liquid chromatography at the temperatures from 298.15 to 368.15 K. The partial molar excess enthalpies at infinite dilution values Delta H 1 (E,infinity) were calculated from the experimental gamma 13 (infinity) values obtained over the temperature range. The selectivities for the hexane/benzene, cyclohexane/benzene, n-hexane/thiophene, n-decane/thiophene, cyclohexane/thiophene, toluene/thiophene, and oct-1-ene/thiophene separation problems were calculated from the gamma 13 (infinity). Obtained values were compared to the literature values for the other ionic liquids, NMP, and sulfolane.

Ensuring the Validity of the Micro Foundation in DSGE Models

The presence of i) stochastic trends, ii) deterministic trends, and/or iii) stochastic volatility in DSGE models may imply that the agents' objective functions attain infinite values. We say that such models do not have a valid micro foundation. The paper derives sufficient conditions which ensure that the objective functions of the households and the firms are finite even when various trends and stochastic volatility are included in a standard DSGE model. Based on these conditions we test the validity of the micro foundation in six DSGE models from the literature. The models of Justiniano & Primiceri (American Economic Review, forthcoming) and Fernandez-Villaverde & Rubio-Ramirez (Review of Economic Studies, 2007) do not satisfy these sufficient conditions, or any other known set of conditions ensuring finite values for the objective functions. Thus, the validity of the micro foundation in these models remains to be established.

Interactions of polyethylene glycols with water studied by measurements of density and sound velocity

Densities and sound velocities of ethylene glycol (EG) and polyethylene glycols (PEGs) of molecular weight 200, 300, 400, 550, 600, 1000, 1450, 3350, 8000, and 10,000 at (288.15, 298.15, and 308.15) K were measured with high precision vibrating tube densimeter and sound velocity measuring device. They were used to evaluate apparent molar volumes, V ∅, and apparent molar isentropic compressibilities, K ΦS . Infinite dilution values of these parameters, V ∅ ∘ , and K Φ S ∘ , were obtained from their plot as a function of molality. The variations of V ∅ ∘ , and K Φ S ∘ , with the number of repeating units in PEGs and with temperature were examined. Comparison of the experimentally obtained data was made with the available literature data and also with some values predicted according to group additivity approach. The results were interpreted in terms of hydration and conformational effects of PEGs in water. A correlation was also examined between V ∅ ∘ or K Φ S ∘ values of PEGs in water and equilibrium moisture contents of PEGs as well as the water vapor permeabilities (WVP) of edible films containing PEGs.

Activity coefficients at infinite dilution measurements for organic solutes in the ionic liquid trihexyltetradecylphosphonium-bis-(2,4,4-trimethylpentyl)-phosphinate using g.l.c. at T = (303.15, 308.15, 313.15, and 318.15) K

A new phosphonium ionic liquid was investigated as a possible solvent for use in the separation of aromatic hydrocarbons from aliphatic hydrocarbons. The activity coefficients at infinite dilution, γ 13 ∞ , for 14 solutes: n-alkanes, alk-1-enes, alk-1-ynes, cycloalkanes, and aromatic hydrocarbons in the ionic liquid trihexyltetradecylphosphonium-bis-(2,4,4-trimethylpentyl)-phosphinate [3C 6C 14P][(C 8H 17) 2PO 2] were consequently determined by gas–liquid chromatography at four temperatures T = (303.15, 308.15, 313.15, and 318.18) K. The partial molar excess enthalpies at infinite dilution values, Δ H 1 E, ∞ , were calculated from experimental γ 13 ∞ values obtained over the temperature range studied. The selectivities for the n-hexane/benzene, hex-1-yne/benzene, and cyclohexane/benzene separations scenarios were calculated from the γ 13 ∞ and were compared to studies from the recent literature for results of other ionic liquids, as well as NMP and sulfolane, taken from the recent literature. The results indicate that [3C 6C 14P][(C 8H 17) 2PO 2] would make a poor solvent for the separation of an alkane or alk-1-ene or cycloalkene from benzene. The results do, however, expand our knowledge and point to some interesting facets of (solute + ionic liquid) interactions.

Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms

The submodular function minimization problem (SFM) is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial-time algorithms have recently been discovered to solve this problem. The most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally-ordered finite set and whose range includes both finite and infinite values. In this paper we demonstrate that this general form of SFM is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints. These tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism. This larger tractable class also includes the problem of satisfying a set of Horn clauses ( Horn-SAT), as well as various extensions of this problem to larger finite domains.

Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects

In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type \({\int_\Omega A_n\nabla u\cdot\nabla u\,dx+\int_\Omega u^2d\mu_n}\), where An is a symmetric positive definite matrix-valued function and μn is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of An we prove that the limit energy belongs to the same class, i.e. its reads as \({\hat F(u)+\int_\Omega u^2d\mu}\), where \({\hat F}\) is a diffusion independent of μn and μ is a nonnegative Borel measure which does depend on \({\hat F}\) . This compactness result extends in dimension two the ones of [11,23] in which An is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear. However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates, the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an explicit formula for the limit energy specifying the kernel of the nonlocal term.

On the regularity of some thermocapillary convection models

The classical model of confined thermocapillary convection is analyzed. Its vorticity singularity, independent of the contact angle, leads to infinite pressure values at contact lines, forbidding any numerical use of the Laplace equation to calculate free surface shapes. Four models are explored to overcome this difficulty: an explicit polynomial filtering, a Navier slip at the solid boundaries, an interface viscosity model and the combination of slip and interface viscosity. Regular solutions are obtained with the first and last approaches. Only the last one is based on physical considerations and, by the introduction of physical length scales, avoids infinite pressure values at the contact line.

Radiation and scattering from imperfect cylindrical electromagnetic cloaks

The design of electromagnetic invisibility cloaks is based on singular mappings prescribing zero or infinite values for material parameters on the inner surface of the cloak. Since this is only approximately feasible, an asymptotic analysis is necessary for a sound description of cloaks. We adopt a simple and effective approach for analyzing electromagnetic cloaks - instead of the originally proposed singular mapping, nonsingular mappings asymptotically approaching the ideal one are considered. Scattering and radiation from this type of imperfect cylindrical cloaks is solved analytically and the results are confirmed by full-wave finite element simulations. Our analysis sheds more light on the influence of this kind of imperfection on the cloaking performance and further explores the physics of cloaking devices.