Single-valued Function Research Articles

Dynamic Investment Strategies: Portfolio Insurance versus Efficient Frontier

Selected dynamic investment strategies are analyzed within a unifying theoretical framework. We suggest a Kolmogorov-type partial differential equation for a profit and loss (P&L) distribution of strategies contingent on the current value of the basic asset as well as on a balance of a trading account ?P&L-to-date. This gives a possibility to study much wider class of strategies than is usually done in the literature and practical applications. Using our equation we build dynamic efficient frontier and demonstrate that an attempt to minimize variance for a given expected profit leads to a contrarian trading strategy, also known as the St.Petersburg paradox. Similar analysis is performed for the Black-Jones-Perold constant proportion portfolio insurance (CPPI). It is shown that both, dynamic efficient frontier and CPPI, belong to a special class of power-option-replicating strategies. Despite its small Sharpe ratio CPPI has an advantage of controlled downside, as its PL distribution is far from gaussian. Conversely, Sharpe-optimal dynamic efficient frontier has an uncontrollable downside. We show how to blend the advantages of these two strategies. A special part is devoted to discrete analysis and risk capital considerations for non-replicating strategies when P&L-to-date is not a single-valued function of the current asset value and path-dependency is essential.

Effects of Size and Slenderness on Ductility of Fracturing Structures

The ductility of an elastic structure with a growing crack may be defined as the ratio of the additional load-point displacement that is caused by the crack at the moment of loss of stability under displacement control to the elastic displacement at no crack at the moment of peak load. The stability loss at displacement control is known to occur when the load-deflection curve of the whole structural system with the loading device (characterized by a spring) reaches a snapback point. Based on the known stress intensity factor as a function of crack length, the well-known method of linear elastic fracture mechanics is used to calculate the load-deflection curve and determine the states of snapback and maximum loads. An example of a notched three-point bend beam with a growing crack is analyzed numerically. The ductility is determined and its dependence of the structure size, slenderness, and stiffness of the loading device is clarified. The family of the curves of ductility versus structure size at various loading device stiffnesses is found to exhibit at a certain critical stiffness a transition from bounded single-valued functions of D to unbounded two-valued functions of D. The method of solution is general and is applicable to cracked structures of any shape. The flexibility (force) method can be adapted to extend the ductility analysis to structural assemblages provided that the stress intensity factor of the cracked structural part considered alone is known. This study leads to an improved understanding of ductility, which should be useful mainly for design against dynamic loads.

Impedance boundary conditions for a metal film with a rough surface

We have obtained local impedance boundary conditions for a metal film characterized by an isotropic, frequency-dependent, complex dielectric function epsilon(omega) that occupies the region zeta(x(1)) < x(3) < D. The surface-profile function zeta(x(1)) is assumed to be a single-valued function of x(1) that is differentiable as many times as is necessary. The electromagnetic field in the system is assumed to be p polarized with the plane of incidence, the x(1)x(3) plane. The results are used to study the scattering of p-polarized light from and its transmission through the metal film when the surface-profile function zeta(x(1)) in these calculations is assumed to be a stationary, zero-mean, Gaussian random process. These calculations are approximately four times faster than rigorous computer simulations, and their results are qualitatively and quantitatively in good agreement with those of the latter simulations.

Initial-interval driven biasing of nodal concentration along a fixed curve

Element size transitioning in the construction of spatial meshes for finite element models is often controlled by biasing the concentration of nodes, towards one end or the other, along each of a set of curves in the model. A simple, common and efficient scheme to implement such nodal concentration biasing along a given curve is to require that the nodal spacings δ i be (sequence) terms b iδ 0 of a geometric series. Current practice takes the parameter value b, or its equivalent, as an independent input, so that the initial nodal spacing δ 0 must be a computed output. This is the most straightforward approach, but the lack of direct control over the value δ 0 is a significant shortcoming. In an element size transitioning scenario, δ 0 is often a parameter for which the model builder/analyst has independent quantitative information. It may represent the a priori known thickness of a thin bond or weld, for example. A more rational choice for these cases, proposed by this paper, is a scheme for which δ 0 is an independent input parameter instead of b. The parameter b is computed by a convergence-guaranteed algorithm for which the existence of b as a single-valued function of its input is proven.

Self-Affine Fractal Characterization of a TNT Fracture Surface

AbstractA trinitrotoluene (TNT) fracture surface image is characterized in terms of a self-affine fractal structure. The fracture surface was produced by high acceleration in an ultracentrifuge when the TNT strength was exceeded. An atomic force microscope (AFM) captured the topography of a 4 micron square region on the fracture surface. The present analysis supports a self-affine description of the TNT fracture surface (wavelengths of 0.016 micron to 4.0 micron) and provides a new prespective on fracture processes in TNT. An essential step in self-affine fractal characterization of surfaces is the determination of reference surfaces. A self-affine fracture surface can be described in terms of a single-valued height function. In the TNT fracture surface, single-valued height functions, which describe surface texture can only be defined with respect to curved reference surfaces. By employing curved reference surfaces, we have demonstrated that self-affine fractal scaling can be used to characterize the TNT fracture surface. This provides important information that is not evident in the analysis of individual surface scans.

On a construction of a hierarchy of best linear spline approximations using repeated bisection

We present a method for the construction of hierarchies of single-valued functions in one, two, and three variables. The input to our method is a coarse decomposition of the compact domain of a function in the form of an interval (univariate case), triangles (bivariate case), or tetrahedra (trivariate case). We compute best linear spline approximations, understood in an integral least squares sense, for functions defined over such triangulations and refine triangulations using repeated bisection. This requires the identification of the interval (triangle, tetrahedron) with largest error and splitting it into two intervals (triangles, tetrahedra). Each bisection step requires the recomputation of all spline coefficients due to the global nature of the best approximation problem. Nevertheless, this can be done efficiently by bisecting multiple intervals (triangles, tetrahedra) in one step and by reducing the bandwidths of the matrices resulting from the normal equations.

Lazy computation with exact real numbers

We provide a semantical framework for exact real arithmetic using linear fractional transformations on the extended real line. We present an extension of PCF with a real type which introduces an eventually breadth-first strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if , rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the one-point compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of single-valued elementary functions.

A continuous plate-tectonic model using geophysical data to estimate plate-margin widths, with a seismicity-based example

SUMMARY A continuous kinematic model of present-day plate motions is developed which (1) provides more realistic models of plate shapes than employed in the original work of Bercovici & Wessel (1994), and (2) provides a means whereby geophysical data on intraplate deformation is used to estimate plate-margin widths for all plates. A given plate’s shape function (which is unity within the plate and zero outside the plate) can be represented by analytic functions as long as the distance from a point inside the plate to the plate’s boundary can be expressed as a single-valued function of azimuth (i.e. a single-valued polar function). To allow su⁄cient realism to the plate boundaries, without the excessive smoothing used by Bercovici & Wessel, the plates are divided along pseudo-boundaries; the boundaries of plate sections are then simple enough to be modelled as single-valued polar functions. Moreover, the pseudo-boundaries have little or no eiect on the ¢nal results. The plate shape function for each plate also includes a plate-margin function which can be constrained by geophysical data on intraplate deformation. We demonstrate how this margin function can be determined by using, as an example data set, the global seismicity distribution for shallow (depths less than 29 km) earthquakes of magnitude greater than 4. Robust estimation techniques are used to determine the width of seismicity distributions along plate boundaries; these widths are then turned into plate-margin functions, that is analytic functions of the azimuthal polar coordinate (the same azimuth of which the distance to the plate boundary is a single-valued function). The model is used to investigate the eiects of ‘realistic’ ¢nite-margin widths on the Earth’s present-day vorticity (i.e. strike-slip shear) and divergence ¢elds as well as the kinetic energies of the toroidal (strike-slip and spin) and poloidal (divergent and convergent) £ow ¢elds. The divergence and vorticity ¢elds are far more well de¢ned than for the standard discontinuous plate model and distinctly show the in£uence of diiuse plate boundaries such as the northeast boundary of the Eurasian plate. The toroidal and poloidal kinetic energies of this model diier only slightly from those of the standard plate model; the diierences, however, are systematic and indicate a greater proportion of spin kinetic energy in the continuous plate model.

Relation of a frequency-dependent structure with the corresponding frequency-independent structure

It is shown that the second theorem in the previous paper [Nakamura and Takewaki (1989) Optimal elastic structures with frequency-dependent elastic supports. International Journal of Solids and Structures25(5), 539–551] can be extended to a more general case where there is no special restriction on the characteristics of frequency-dependent stiffnesses except their positive definiteness and a characteristic as a single-valued function. This theorem enables one to disclose the relationship between the design space of elastic structures with frequency-dependent stiffnesses and that of elastic structures with the corresponding frequency-independent stiffnesses, both with respect to fundamental natural frequency. It facilitates not only clarification of lowest-mode qualification conditions for the frequency-dependent model with the help of the frequency-independent model, but also use of the modal analysis technique via the substitute model (the frequency-independent model).

Diffraction by a Resistive Sheet Attached to a Two-sided Impedance Plane

ABSTRACT The diffraction by a resistive sheet attached to a two-sided impedance plane, made up by perfectly electric conducting and impedance half-planes, is presented. An E-plane wave normally illuminates this structure, therefore, the problem is a two dimensional one. By using Sommerfeld-Maliuzhinets' method, the problem is reduced to the solution of a coupled system of functional equations for two spectral functions corresponding to the two spatial regions defined by the resistive sheet. By eliminating either of the spectral functions, a second-order difference equation with variable 2π-periodic coefficients is obtained for the remaining one. A general method of constructing a single-valued solution of this second-order difference equation is presented based on the Fourier transform. It is shown that the obtained single-valued meromorphic spectral function satisfies the edge condition, pole requirement and the radiation condition.

LOW-GRAVITY PROPELLANT SLOSH ANALYSIS USING SPHERICAL COORDINATES

Low-gravity sloshing in a convex axisymmetrical container subjected to lateral excitation is formulated by a variational principle and is solved analytically by a modal analysis method. The use of spherical coordinates enables us: (i) to determine analytically the system of characteristic functions for an arbitrary convex container, for which time-consuming and expensive numerical methods have been used in the past; (ii) to express the liquid surface and its dynamical displacement in the form of a single-valued function even when the liquid surface curves strongly, due to surface tension; and (iii) to satisfy the compatibility condition for the liquid surface displacement at the container wall. The variational principle is transformed into a frequency equation in the form of a standard eigenvalue problem by the Galerkin method, in which admissible functions for the velocity potential and the liquid surface displacement are determined analytically in terms of the Gauss hypergeometric series. Since this process is analytical and the dimension of the eigenvalue problem for obtaining a sufficiently converged solution is very low, little computation time and cost are involved. Numerical results show that: (a) the influences of the Bond number on the eigenfrequency are different for high and low liquid-filling levels; (b) neglecting the surface tension underestimates the magnitude of the surface oscillation; and (c) the liquid depth yielding the maximum slosh force and moment increases with decreasing Bond number.

On parallel hierarchies and Rki

This paper defines natural hierarchies of function and relation classes □ i, k c and Δ i, k c , constructed from parallel complexity classes in a manner analogous to the polynomial-time hierarchy. It is easily shown that □ i−1, k p ⊆ □ c, k c ⊆ □ i, k p and similarly for the Δ classes. The class □ i,3 c coincides with the single-valued functions in Buss et al.'s class F℘ 3 Σ i−1,3 p [wit, log O(1)] (see [11]), and analogously for other growth rates. Furthermore, the class □ i, k c comprises exactly the functions Σ i, k b -definable in S k i−1 , and if T k i−1 is ∀∃ Σ i, k b -conservative over S k i−1 , then □ i, k p is completely parallelizable. All functions in □ i, k c are Σ i, k b -definable in R k i ; this suffices to show that if the known ∀∃ Σ i, k b conservativity between R 3 i and S 3 i−1 extends to R 2 i and S 2 i−1 , then NC = NC 1 relative to an oracle in PH. We prove a KPT-style witnessing theorem for S k i using constantly many rounds of □ i, k c interactive computation, and thus show that if S k i ≡ R k i+1 then the bounded arithmetic hierarchy collapses, provably in S k i .

The Model of Hysteretic Behavior of SMA Based on the High Order Approximation of Differential Equation Method

Due to thermomechanical hysteresis in solids undergoing martensitic type of structure transformations the macroscopic strain and volume fraction of martensite are not a single-valued function of stress and temperature, but they become functions of the process of their change. Some of the phenomenological approaches describing main thermomechanical properties of shape-memory alloys (SMA) connected with hysteresis were recently developed. A special type of differential equations (DE) describing evolution of the inelastic macroscopic strain and volume fraction of martensite as functions of the temperature have been proposed in out recent papers. Simplest applications of these equations to a strain evolution during the multiple temperature cycling in a small temperature interval live been also discussed. Some other problems associated with the irreversible processes caused by hysteresis will be discussed in the present paper. In particular, a new interpretation of DE method based on the transition probability concept and high order approximation inciuding the return point effect is considered.

Erratum: “Energy switching approach to potential surfaces: An accurate single-valued function for the water molecule” [J. Chem. Phys. 105, 3524 (1996)

Erratum: “Energy switching approach to potential surfaces: An accurate single-valued function for the water molecule” [J. Chem. Phys. <b>105</b>, 3524 (1996)

Numerical simulation of three-dimensional free surface flow in isopycnal co-ordinates

In this paper a semi-implicit method for three-dimensional circulation in isopycnal co-ordinates is derived and discussed. It is assumed that the flow is hydrostatic and characterized by isopycnal surfaces which can be represented by explicit, single-valued functions. The hydrostatic pressure is determined by using the conjugate gradient method to solve a block pentadiagonal linear system. The horizontal velocities are determined by solving a large set of tridiagonal systems. The stability of the resulting algorithm is shown to be independent of the surface and internal gravity wave speeds.

Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces and BMO

The boundedness of the Hardy operator and the Hardy-Littlewood operator are established, respectively, in and the space BMO of functions of bounded mean oscillation on the real axis . Here the space is isomorphic to the Hardy space of single-valued analytic functions in the upper half-plane satisfying condition (0.3), the Hardy-Littlewood operator is defined in by equality (0.2), and the Hardy operator is defined in by equality (0.1) and its value is continued to as an even (odd) function if the function is even (odd). For an arbitrary function one sets , where is the even and is the odd component of .

Thermodynamics and kinetics of adsorption of cyanide ions on silver. Part I

The adsorption of cyanide ions on silver has been studied by measuring the electrode impedance. It has been found that in the region of potentials E < −0.5 V (SHE), the process under study involves the stage at which the substance is supplied to the electrode and the adsorption-desorption stage itself. At more positive potentials, the process becomes more complicated. The capacity of the electric double layer and the potential of zero charge have been determined. It has been established that in the description of the adsorption of cyanide ions on silver, the potential of the electrode E rather than its charge should be used. At E < −0.5 V (SHE), the adsorption is represented by the Parsons isotherm, and at E > −0.5 V (SHE) by the virial isotherm. The structure of the electric double layer is considered. The components of the integral capacity of the compact layer are calculated as single-valued functions of the electrode potential. The standard Gibbs adsorption energy and its electrostatic and non-coulomb components are calculated.

Light scattering by a reentrant fractal surface

Recently, rigorous numerical techniques for treating light scattering problems with one-dimensional rough surfaces have been developed. In their usual formulation, these techniques are based on the solution of two coupled integral equations and are applicable only to surfaces whose profiles can be described by single-valued functions of a coordinate in the mean plane of the surface. In this paper we extend the applicability of the integral equation method to surfaces with multivalued profiles. A procedure for finding a parametric description of a given profile is described, and the scattering equations are established within the framework of this formalism. We then present some results of light scattering from a sequence of one-dimensional flat surfaces with defects in the form of triadic Koch curves. Beyond a certain order of the prefractal, the scattering patterns become stationary (within the numerical accuracy of the method). It can then be argued that the results obtained correspond to a surface with a fractal structure. These constitute, to our knowledge, the first rigorous calculations of light scattering from a reentrant fractal surface.

Light velocity in nonrelativistic quantum mechanics on a circle

A single-valued Schrodinger wave function on a circle is discussed in the framework of the path integral. It is clarified how the light velocity comes into play and how terms corresponding to superluminal transmission of signals are suppressed.

Density and finiteness. A discrete approach to shape

We show in this paper that the category of shape can be modelled in discrete terms using maps defined in dense subsets of compacta. This approach to shape provides in addition a characterization of the shape of compacta which does not require external elements and uses only continuous single-valued functions, in contrast with the existing internal characterizations of shape. As an application, we prove a connection between the notion of shape image, due to Lisica, and the basic notion of omega limit of a dynamical system.