Single-valued Function Research Articles

The projective translation equation and unramified 2-dimensional flows with rational vector fields

Let x = (x, y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1−z)ϕ(x) = ϕ(ϕ(xz)(1−z)/z); here ϕ(x) = (u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in \({\mathbb{C}^{2}\setminus\{\text{union of curves}\}}\)) projective flows whose vector field is still rational. If an orbit of the flow is given by a homogeneous rational function of degree N, then N is called the level of the flow. We prove that, up to conjugation with 1-homogenic birational plane transformation, unramified non-singular flows are of 6 types: (1) the identity flow; (2) one flow for each non-negative integer N—these flows are rational of level N; (3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; (4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; (5) the level 4 flow expressable in terms of lemniscatic elliptic functions; (6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Pólya–Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus. This study, though seemingly analytic, is in fact algebraic and our method reduces to algebraic transformations in various quotient rings of rational function fields.

The characterization of some set-valued iteration semigroups

We give the characterization of set-valued iteration semigroups which are the counterparts of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.

Catalytic consequences of the identity and coverages of reactive intermediates during methanol partial oxidation on Pt clusters

Rate measurements, carried out after elimination of transport corruptions, show that CH3OH partial oxidation on Pt clusters proceeds via three distinctly different kinetic regimes defined by the operating oxidant-to-reductant (O2-to-CH3OH) ratio. These kinetic regimes, each with unique rate dependencies for HCHO, HCOOCH3, and CO2 formation, arise from a dynamic shift in the identity and coverages of reactive intermediates on Pt cluster surfaces from saturated with chemisorbed oxygen, to uncovered, and then to partially covered with methanol derived species as the oxygen chemical potential at cluster surfaces decreases with decreasing O2-to-CH3OH ratio. HCHO formation rates first remain insensitive to O2 on Pt cluster surfaces saturated with chemisorbed oxygen (O*), then increase proportionally with O2 as cluster surfaces depleted of oxygen and as unoccupied Pt sites become the most abundant surface intermediates, and eventually increase more than linearly with O2 on Pt surfaces predominantly covered with CH3OH derived intermediates. The transition in rate dependence on O2 is accompanied by a concomitant transition on CH3OH dependence: HCHO formation rates are linearly proportional to, then independent of, and eventually inhibited by CH3OH as the O2-to-CH3OH ratio decreases. Similar transition in reaction orders is also found for HCOOCH3 and CO2 formation. The first-order rate coefficient for CH3OH oxidation is a single-valued function of the operating O2-to-CH3OH ratio, because the ratio determines the oxygen chemical potential at Pt cluster surfaces via the kinetic coupling of O2 and CH3OH activation steps and, in turn, the identity and coverage of reactive intermediates, the nature and availability of active sites, and the identity of kinetically-relevant steps.

Visualisation of complex functions on Riemann sphere

The notion of a multi-valued function is frequent in complex analysis and related fields. A graph of such a function helps to inspect the function, however, the methods working with single-valued functions can not be applied directly. To visualize such a type of function, its Riemann surface is often used as a domain of the function. On such a surface, a multi-valued function behaves like a single-valued function. In our paper, we give a quick overview of the proposed method of visualization of a single-valued complex function over its Riemann sphere. Then, we pass to the adaptation of this method on the visualization of a multi-valued complex function. Our method uses absolute value and argument of the function to create the graph in 3D space over the Riemann sphere. Such a graph provides an overview of the function behavior over its whole domain, the amount and the position of its branch points, as well as poles and zeros and their multiplicity. We have also created an algorithm for adaptive grid which provides higher density of vertices in areas with higher curvature of the graph. The algorithm eliminates the alias in places where the branches are joined together.

Survival of radiation-damaged cells via mechanism of repair by pool molecules: the Lambert function as the exact analytical solution of coupled kinetic equations

Repair of radiation damaged cells can be carried out through their interactions with intracellular substances that can supply the needed energy for repair. These substances may be viewed as forming a pool of repair molecules that through chemical reactions with lesions can lead to cell recovery from the initial radiation insult by deposition of dose $$D$$ . Presently, time evolution of mean concentrations of interacting substances is obtained by solving the corresponding rate equations given by a coupled system of second-order non-linear differential equations that are imposed by the mass action law. For cell surviving fractions after irradiation, the most important quantity is the time-dependent concentration of lethal lesions. Our main working hypothesis is that pool substances are capable of repairing the inflicted injury to any cell molecules, including deoxyribonucleic acid which is generally viewed as the most critical target of radiation. The previous solution of these rate equations is only formal as it is expressed by yet another equation of an implicit, transcendental form. In the earlier applications, this formal solution has only been tackled by numerical means that, however, have no connection with any of the myriad of the usual explicit forms of cell surviving fractions. This drawback effectively discouraged researchers from further explorations of the otherwise attractive pool methodology. Such a circumstance is unfortunate in light of a clear and advantageous radiobiological interpretation of the parameters of this theoretical formalism of chemical kinetics.The present study is aimed at rescuing the pool methodology by solving the underlying transcendental equation for lethal lesions uniquely, exactly and explicitly in terms of the principal value Lambert $$W_0$$ function. This is a single-valued and dose-dependent function, which can be readily and accurately computed either from the available fast numerical algorithms or by employing the existing simple closed expressions with a quotient of elementary, logarithmic functions. Another distinct advantage of this analytical result is the known behaviors of $$W_0$$ at small and large doses. This permits an easy and immediate identification of the final $$D_0$$ (or $$D_{37}$$ ) dose and the extrapolation number $$n$$ . Such a circumstance offers new possibilities within the presently proposed “Pool Repair Lambert” (PRL) model for analysis of problems encountered in assessing cell survival after exposure to various modalities of radiation, including different schedules (acute, fractionated) for the same radiation quality. Importantly, the PRL model is universally applicable to all doses with a smooth passage from low through intermediate to high doses. As to applications in radiotherapy, this feature is particularly important for treatment schedules with high-doses per fraction as in stereotactic radiosurgery and stereotactic body radiotherapy.

Self-Contradictions from the Excessive Use of Natural Units

The hypothesis c = h = G = 1 implies that unit mass is not a single-valued function but rather has two widely varying values, such as 7.4 × 10-51 kg and 4.0 × 1035 kg. Hence, the considerable body of work in theoretical physics that uses this common convention must be deemed suspect. In order to avoid this problem, theoreticians must limit themselves to c = h = 1 or, exclusively, c = G = 1 depending upon whether they are chiefly concerned with atomic physics or with gravity, respectively.

Graphical functions and single-valued multiple polylogarithms

Graphical functions are single-valued complex functions which arise from Feynman amplitudes. We study their properties and use their connection to multiple polylogarithms to calculate Feynman periods. For the zig-zag and two more families of phi^4 periods we give exact results modulo products. These periods are proved to be expressible as integer linear combinations of single-valued multiple polylogarithms evaluated at one. For the larger family of 'constructible' graphs we give an algorithm that allows one to calculate their periods by computer algebra. The theory of graphical functions is used to prove the zig-zag conjecture.

Improved Sobolev Embedding Theorems for Vector-Valued Functions

The aim of this paper is to give an extension of the improved Sobolev embedding theorem for single-valued functions to the case of vector-valued functions which is involved with the three-dimensional massless Dirac operator together with the three- or two-dimensional Weyl-Dirac (or Pauli) operator, the Cauchy-Riemann operator and also the four-dimensional Euclidian Dirac operator.

Non-parametric convex identification of extended generalized Prandtl–Ishlinskii models

The Generalized Prandtl–Ishlinskii model (GPI) of hysteresis has wide applicability, partly because of its capability to model asymmetric hysteresis. It is characterized by three unknown functions. Today, GPI models are typically identified through trial and error by ad hoc methods, presuming parameterized expressions for these functions and then using nonlinear least squares to determine the parameters, with concurrent problems of convergence, a dependence on the initial parameter guess, and local minima. Except for the aggregated hysteresis input–output fit the result gives no information as to whether the functions chosen are appropriate or not. Here we present a method to circumvent these problems for a more general class of hysteresis models. First, we introduce an extended GPI model (XGPI), where an additional memoryless function is placed in parallel to the GPI model. This further widens the applicability, allowing, for example, arbitrary orientation of the hysteresis loop. For such models it is shown how its four separate mappings can be identified by convex optimization. Appropriate single-valued functions can then be fitted individually to the resulting mappings and, if necessary, the function parameters found can be fine-tuned using nonlinear least squares on input–output data. The method is applied to simulated data and experimental data from a magnetoelastic torque sensor, and the results are favorably compared to the results of another commonly used hysteresis model.

Effect of chain flexibility on master curve behavior for diffusion coefficient

The diffusion coefficients of simple chain models are analyzed as a function of packing fraction, η, and as a function of a parameter C that is the density raised to a power divided by temperature to look at scalar metrics to find master curves. The central feature in the analysis is the mapping onto an effective hard site diameter, d. For the molecular models lacking restrictions on dihedral angle (e.g., freely jointed), simple mappings of molecular potential to d work very well, and the reduced diffusion coefficient, D*, collapses into a single-valued function of η. Although this does not work for the dihedral angle restriction case, assuming that d is inversely proportional to temperature to a power results in collapse behavior for an empirically selected value of the power. This is equivalent to D* being a single-valued function of C. The diffusion coefficient of a single-site penetrant in the chain systems also is found to be a scalar metric that can reduce the chain diffusion data for a given system to a single master curve.

Development of Physical-Parameter Identification Procedure for Energy-Dissipated Buildings with Symmetric Ductile Braces

In this paper, a pseudo-single-degree-of-freedom system identification procedure is developed to investigate the dynamic characteristics of energy-dissipated buildings equipped with symmetric ductile braces (SDBs). The primary structure is assumed to be linear on account of substantial reduction of seismic forces due to the installation of SDBs for which a bilinear hysteretic model is considered. The hysteretic model is in turn characterized by a backbone curve by which the multi-valued restoring force is transformed into a single-valued function. With the introduction of backbone curves, the system identification analysis of inelastic structures is significantly simplified. The proposed algorithm extracts individually the physical parameters of each primary structure and each energy-dissipation device that are considered useful information in the structural health monitoring. A numerical example is conducted to demonstrate the feasibility of using the proposed technique for physical parameter identification of partially inelastic energy-dissipated buildings.

Analytic structure of the multichannel Jost matrix for potentials with Coulombic tails

A quantum system is considered that can move in N two-body channels with the potentials that may include the Coulomb interaction. For this system, the Jost matrix is constructed in such a way that all its dependencies on the channel momenta and Sommerfeld parameters are factorized in the form of explicit analytic expressions. It is shown that the two remaining unknown matrices are single-valued analytic functions of the energy and therefore can be expanded in the Taylor series near an arbitrary point within the domain of their analyticity. It is derived a system of first-order differential equations whose solutions determine the expansion coefficients of these series. Alternatively, the unknown expansion coefficients can be used as fitting parameters for parametrizing experimental data similarly to the effective-range expansion. Such a parametrization has the advantage of preserving proper analytic structure of the Jost matrix and can be done not only near the threshold energies, but around any collision or even complex energy. As soon as the parameters are obtained, the Jost matrix (and therefore the S-matrix) is known analytically on all sheets of the Riemann surface, and thus enables one to locate possible resonances.

Three Fingerprints of Memristor

This paper illustrates that for a device to be a memristor it should exhibit three characteristic fingerprints: 1) When driven by a bipolar periodic signal the device must exhibit a “pinched hysteresis loop” in the voltage-current plane, assuming the response is periodic. 2) Starting from some critical frequency, the hysteresis lobe area should decrease monotonically as the excitation frequency increases, and 3) the pinched hysteresis loop should shrink to a single-valued function when the frequency tends to infinity. Examples of memristors exhibiting these three fingerprints, along with non-memristors exhibiting only a subset of these fingerprints are also presented. In addition, two different types of pinched hysteresis loops; the transversal (self-crossing) and the non-transversal (tangential) loops exhibited by memristors are also discussed with its identification criterion.

Real-valued Choquet integrals for set-valued mappings

In this paper a new kind of real-valued Choquet integrals for set-valued mappings is introduced, and some elementary properties of this kind of Choquet integrals are studied. Convergence theorems of a sequence of Choquet integrals for set-valued mappings are shown. However, in the case of the monotone convergence theorem of the nonincreasing sequence of Choquet integrals for set-valued mappings, we point out that the integrands must be closed. Specially, this kind of real-valued Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.

Elastic moduli and poisson ratio for amorphous polymers and glasses

We propose that the product of the density by the square of the rms velocity of strain waves, which exhibits the features typical of elastic moduli, be referred to as effective (or characteristic) elastic modulus. The ratio of the bulk compression modulus to the effective elastic modulus is a single-valued function of the Poisson ratio not only for crystals and glasses, but also for amorphous organic polymers. The effective elastic modulus may be helpful in analysis of anharmonism of lattice vibrations of deformable bodies.

Stakeholder Happiness Enhancement: A Neo-Utilitarian Objective for the Modern Corporation

ABSTRACT:Employing utilitarian criteria, Jones and Felps, in “Shareholder Wealth Maximization and Social Welfare: A Utilitarian Critique” (Business Ethics Quarterly23[2]: 207–38), examined the sequential logic leading from shareholder wealth maximization to maximal social welfare and uncovered several serious empirical and conceptual shortcomings. After rendering shareholder wealth maximization seriously compromised as an objective for corporate operations, they provided a set of criteria regarding what a replacement corporate objective would look like, but do not offer a specific alternative. In this article, we draw on neo-utilitarian thought to advance a refined version of normative stakeholder theory that we believe addresses a major remaining criticism of extant versions, their lack of specificity. More particularly, we provide a single-valued objective function for the corporation—stakeholder happiness enhancement—that would allow managers to make principled choices between/among policy options when stakeholder interests conflict.

Scattering of a surface plasmon polariton by a localized dielectric surface defect

On the basis of a rigorous, nonperturbative, purely numerical solution of the corresponding reduced Rayleigh equation for the scattering amplitudes we have studied the scattering of a surface plasmon polariton by a two dimensional dielectric defect on a planar metal surface. The profile of the defect is assumed to be an arbitrary single-valued function of the coordinates in the plane of the metal surface, and to be differentiable with respect to those coordinates. When the defect is circularly symmetric and the dependence of the scattering amplitudes on the azimuthal angle is expressed by a rotational expansion, the reduced Rayleigh equation is transformed into a pair of one-dimensional integral equations for each value of the rotational quantum number. This approach is applied to a defect in the form of an isotropic Gaussian function. The differential cross sections for the scattering of the incident surface plasmon polariton into volume electromagnetic waves in the vacuum above the surface and into other surface plasmon polaritons are calculated, as well as the intensity of the field near the surface. These results differ significantly from the corresponding results for a metallic defect on a metallic substrate.

Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions

This paper is concerned with nonlinear nonlocal differential inclusion of evolution type in Fréchet spaces, defined on right half-line. The underlying feature of the inclusion under consideration is that it is non-autonomous. We obtain some compactness characterizations of integral solution sets for the inclusion without nonlinear perturbations. Then, making use of these characterizations, we derive a new existence result of global integral solutions for the original inclusion. No invariance condition on the nonlinearity is involved. The results we obtained here extend the semilinear case of the previous related ones such as [I.I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012) 1363–1391] and are new even for the case of the nonlinearity being a single-valued function.

Optimal reconstruction of the folding landscape using differential energy surface analysis

In experiments and in simulations, the free energy of a state of a system can be determined from the probability that the state is occupied. However, it is often necessary to impose a biasing potential on the system so that high energy states are sampled with sufficient frequency. The unbiased energy is typically obtained from the data using the weighted histogram analysis method (WHAM). Here we present differential energy surface analysis (DESA), in which the gradient of the energy surface, dE/dx, is extracted from data taken with a series of harmonic biasing potentials. It is shown that DESA produces a maximum likelihood estimate of the folding landscape gradient. DESA is demonstrated by analyzing data from a simulated system as well as data from a single-molecule unfolding experiment in which the end-to-end distance of a DNA hairpin is measured. It is shown that the energy surface obtained from DESA is indistinguishable from the energy surface obtained when WHAM is applied to the same data. Two criteria are defined which indicate whether the DESA results are self-consistent. It is found that these criteria can detect a situation where the energy is not a single-valued function of the measured reaction coordinate. The criteria were found to be satisfied for the experimental data analyzed, confirming that end-to-end distance is a good reaction coordinate for the experimental system. The combination of DESA and the optical trap assay in which a structure is disrupted under harmonic constraint facilitates an extremely accurate measurement of the folding energy surface.

On Two Aspects of the Painlevé Analysis

We use the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair.