Martin Formula Research Articles

A representation formula for transition probability densities of diffusions and applications

We establish a representation formula for the transition probability density of a diffusion perturbed by a vector field, which takes a form of Cameron–Martin's formula for pinned diffusions. As an application, by carefully estimating the mixed moments of a Gaussian process, we deduce explicit, strong lower and upper estimates for the transition probability function of Brownian motion with drift of linear growth.

A comparison of bedside renal function estimates and measured glomerular filtration rate (Tc99mDTPA clearance) incancer patients

BackgroundThe aim of this study was to compare measured glomerular filtration rate (GFR) with estimates of GFR derived from the population pharmacokinetic methods of Martin and Wright, and the creatinine clearance (CrCl) estimates of Cockcroft and Gault, and Jelliffe. Patients and methodsGFR was determined by technetium-99m diethyl triamine penta-acetic acid (Tc99mDTPA) clearance in adult cancer patients. Height, actual body weight and serum creatinine were measured, and GFR and CrCl estimates calculated. ResultsOne hundred and twenty-two patients were included. The mean measured GFR was 87 ml/min (range 30–174 ml/min). The mean bias (mean percentage error) was 2, 1, –10 and –17%, and the mean precision (mean absolute percentage error) was 18, 19, 21 and 23% for the Wright, Martin, Cockcroft and Gault, and Jelliffe formulas, respectively. The Martin formula significantly underestimates GFR for females (mean bias –10%) and overestimates GFR for males (mean bias 8%) (P <0.001 for bias of males versus females). The Wright and Martin formulas significantly overestimate GFR <50 ml/min (mean bias 39 and 30%; P = 0.03 and 0.05, respectively) and all formulas underestimate GFR >100 ml/min (mean bias –18, –16, –24 and –32% for Wright, Martin, Cockcroft and Gault, and Jelliffe formulas, respectively; P <0.001). ConclusionsAll the assessed estimates for renal function were found to have significant limitations.

A Generalized Cameron–Martin Formula with Applications to Partially Observed Dynamic Portfolio Optimization

The optimal dynamic allocation problem for a Bayesian investor is addressed when the stock's drift—modeled as a linear mean‐reverting diffusion—is not observed directly but only via the measurement process. Adopting a martingale approach, an appropriate generalization of the Cameron–Martin (1945) formula then enables computation of both the optimal dynamic allocation and the value function for a general utility function, in terms of an inverse Laplace transform of an explicit expression. Moreover, closed‐form formulas are provided in the case of power utility.

MAXIMIZING THE PROBABILITY OF ACHIEVING A GOAL IN THE CASE OF A PARTIALLY OBSERVED DRIFT PROCESS

We consider an investor with initial wealth X0&lt;1, who wishes to maximize the probability of achieving a goal, XT = 1, when the stock's drift — modeled as a linear mean-reverting diffusion — is not observed directly but only via the measurement process. Adopting a martingale approach, a generalized Cameron–Martin (1945) formula then enables explicit computation of the value of the problem as well as the wealth process. The dynamic optimal allocation can then be determined using Clark's formula.

Analysis of the bound odd-parity spectrum of krypton by weakest bound electron potential model theory

Within the weakest bound electron potential model theory, the J=0,1,2,3,4,5 odd bound levels of neutral krypton are classified by spectrum-level-like series. In our previous work, we employed Martin's formula to calculate the quantum defects for the unperturbed spectrum-level-like series and obtained excellent results. For the perturbed series, however, extra- or interpolation using Martin's formula may lead to serious error. We, therefore, propose a quantum defect formula to fit the perturbed series by combining Martin’s formula with Langer's. The regularity of spectrum-level-like series is investigated with the revised Martin formula. Great agreement with experimental values is obtained, and the deviations between the calculated values and experimental ones are less than 1 cm −1. Meanwhile, some unknown level positions are predicted.

Multiple diffraction model for proton-proton elastic scattering and total cross section extrapolations to cosmic-ray energies

We analyse pp elastic scattering data at the highest accelerator energy region (between 10 and 63 GeV) through a multiple diffraction approach. The use of Martin's formula in a model developed earlier is substituted by the introduction of a complex elementary (parton-parton) amplitude. With this the total cross section and the real part of the hadronic amplitude may be simultaneously investigated and, with the exception of the diffraction minimum at some ISR energies, a satisfactory description of all experimental data is obtained. Total cross sections extrapolations to cosmic-ray energies (above 6 TeV) show agreement with the reanalysis of the Akeno data performed by N. N. Nikolaev and also with Gaisser, Sukhatme and Yodh results, leading to the prediction 147 mb for the total cross section at 16 TeV. Physical interpretations and critical remarks concerning our parametrizations and results are also presented and discussed.

Comment on the Kundrát-Lokajíček assertion about the applicability of the Martin formula

Contrary to the assertion of Kundr\'at and Lokaj\'{\i}\ifmmode \check{c}\else \v{c}\fi{}ek that the Martin formula, which relates the real part of the elastic hadron-hadron scattering amplitude with its imaginary part, can be used only at small momentum transfers $|t|\ensuremath{\lesssim}0.15(\mathrm{GeV}{/c)}^{2}$ in the CERN ISR and S$p\mathrm{p\ifmmode\bar\else\textasciimacron\fi{}}$S region, the formula can be used consistently at high energies for the observed experimental data in the whole region of the momentum transfer of the experiments. Their errors are due to an inappropriate numerical approach for solving the problem.

On gradients of approximate travelling waves for generalised KPP equations

In this paper we use stochastic semiclassical analysis and the logarithmic transformation to study the gradients of the approximate travelling wave solutions for the generalised KPP equations with Gaussian and Dirac delta initial distributions. We apply the logarithmic transformation to the nonlinear reaction diffusion equations and obtain a Maruyama–Girsanov–Cameron–Martin formula for the driftμ2∇loguμ,uμbeing a solution of a generalised KPP equation. We obtain thatμ2|∇ loguμ(t,x)| is bounded and the trough is flat. The difficult problem in this paper is to prove that the corresponding crest is flat. A probabilistic approach is used in this paper to treat this problem successfully.

Elastic constants of III - V compound semiconductors: modification of Keyes' relation

We have proposed new empirical relations between elastic constants and lattice constants for III - V compound semiconductors with the zincblende (ZB) structure by modifying Keyes' relation. The new relations show excellent agreement with available experimental data and give considerably better estimates than do the extended Hückel tight-binding method and the bond-orbital model. We have calculated the elastic constants for AlP, BAs, AlN, GaN and InN with the ZB structure using the relations. For III nitrides the elastic constants in the wurtzite phase have also been calculated combining the relations and Martin's formula. The values so obtained are reasonable but less reliable than those for the ZB phase.

Feynman integrals and Fredholm determinants

In this article simple properties of Feynman integrals such as the translation invariance, Fubini theorem, and Cameron–Martin formula are used in order to derive a product formula for Fredholm determinants. This formula enables us to compute explicitly some of such determinants as well as the index of the related Fredholm operators. Some examples are done which have applications in the semiclassical approximation (ℏ → 0).

An extension of the Cameron–Martin result

The well-known Cameron–Martin formula allows us to calculate the mathematical expectation where Ws is a Wiener process. This paper extends this result to the case of piecewise continuous martingales. As a particular case the mathematical expectations of a functional of generalized Ornstein– Uhlenbeck processes and pure jump processes are calculated.

An extension of the Cameron–Martin result

The well-known Cameron–Martin formula allows us to calculate the mathematical expectation where Ws is a Wiener process. This paper extends this result to the case of piecewise continuous martingales. As a particular case the mathematical expectations of a functional of generalized Ornstein– Uhlenbeck processes and pure jump processes are calculated.

On the Use of Filtering Theory for the Inversion of Temporal Chemical Systems: Mathematical Formulation

This paper presents a method for inverting experimental data from chemical models by first assuming that the experimental error can be represented as a white noise process and subsequently applying filtering methods to construct an estimator for the unknown parameters. The observations then become time dependent stochastic variables and a particular measurement is equivalent to a realization of this stochastic process. Using some results from filtering theory we can obtain analytically an optimal estimator for the initial conditions and the unknown parameters if the chemical system does not contain random elements. An important tool is assessing the efficiency of the estimator and, therefore, the effectiveness of an experiment is the associated Rao–Cramer lower bound which also can be calculated analytically for noise free models. We show an alternative proof for the Rao–Cramer inequality and establish the relationship between the Rao–Cramer lower bound matrix and the sensitivity analysis of the model. Finally, in the Appendix we derive an extension of the Cameron–Martin formula which is applied to an example in the text.

Analysis of generalized dependences of the viscosity of dilute polymer solutions

Methods and a set of computer programs were developed, in order to identify and discriminate between models describing the concentration dependence of viscosity. The difference was established between competing models for specific systems: poly- p-phenylene terephthalamide-sulphuric acid (96·3 and 99·3%). Most reliable results of predictions were obtained using the Martin formula and the Budtov model. Constants of these models were evaluated. These formulae may be mutually transformed by functional expansion, correct up to high orders of magnitude.

The Feynman maps and the Wiener integral

By introducing the family of Feynman maps ℱs, we show that our earlier definition of the Feynman path integral ℱ=ℱ1 can be obtained as the analytic continuation of the Wiener integral E =ℱ−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps ℱs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for ℱ1=ℱ.

A comment on Martin's relation

In the limit of slowly varying synaptic conductance changes, a quantity proportional to the quantal content of a synaptic potential is provided by Martin's formula: 1/(VO/Vp-1), where Vp is the peak amplitude of the synaptic potential and Vo is the synaptic equilibrium potential. If the synaptic conductance change is not slowly varying on the time scale of the postsynaptic cell's membrane time constant, Martin's formula overestimates the effects of nonlinear summation and provides an upper limit for the actual quantal content. In the limit of rapidly varying synaptic conductance changes the quantity ln(1/[1-Vp/Vo]) is proportional to the quantal content. This formula underestimates the effects of nonlinear summation and gives a lower limit for the quantal content for cases in which the synaptic conductance is not rapidly varying. These two formulas used together provide upper and lower bounds to the correction for nonlinear summation of postsynaptic potentials.