# Journal articles and technical reports

This page lists academic articles. See also informal articles on software development and math and statistics.

John D. Cook, Robert Primmer, Ab de Kwant. Comparing cost and performance of replication in erasure coding. Hitachi Review, vol 63 (July 2014).

*Abstract*. Data storage systems are more reliable than their individual
components. In order to build highly reliable systems out of less reliable
parts, systems introduce redundancy. In replicated systems, objects are simply
copied several times with each copy residing on a different physical device.
While such an approach is simple and direct, more elaborate approaches
such as erasure coding can achieve equivalent levels of data protection
while using less redundancy. This report examines the trade-offs in cost
and performance between replicated and erasure encoded storage systems.

John D. Cook Approximating random inequalities with Edgeworth expansions (2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 78.

*Abstract*. Random inequalities of the form Prob (X > Y + δ) often appear as part of Bayesian clinical trial methods.
Simulating trial designs could require calculating millions of random inequalities.
When these inequalities require numerical integration, or worse random sampling, the inequality calculations account for
the large majority of the simulation time. In this paper we show how to approximate random inequalities using Edgeworth expansions.
The calculations required to use these expansions can be done in closed form, as we will see below.
Although the calculations are elementary, they are also somewhat tedious, and so we include Python code
to illustrate how to use the approximations in practice. We make no distributional assumptions on the random variables
X and Y other than requiring that the necessary moments exist. The accuracy of the approximation will depend on
how well the densities of these random variables are approximated by the Edgeworth expansions.

John D. Cook Fast approximation of gamma inequalities (2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 77.

*Abstract*. Approximation for computing P(*X *> *Y* + δ) for independent gamma random variables *X* and *Y*.

John D. Cook Fast approximation of beta inequalities (2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 76.

*Abstract*. Approximation for computing P(*X *> *Y* + δ) for independent beta random variables *X* and *Y*.

John D. Cook CRM: Prior means and medians (2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 73.

*Abstract*. Resolving the confusion between two ways of specifying a CRM dose-finding trial design.

John D. Cook Random inequalities between survival and uniform distributions (2011). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 71.

*Abstract*.
This note will look at ways of computing P(X>Y) where X is a distribution modeling survival (gamma, inverse gamma, Weibull, log-normal) and Y has a uniform distribution. Each of these can be computer in closed form in terms of common statistical functions. We begin with analytical calculations and then include software implementations in R to make some of the details more explicit. Finally, we give a suggestion for using simulation to compute random inequalities that cannot be computed in closed form.

John D. Cook Basic properties of the soft maximum (2011). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 70.

*Abstract*.
This note presents the basic properties of the soft maximum, a smooth approximation to the maximum of two real variables. It concludes by looking at potential numerical difficulties with the soft maximum and how to avoid these difficulties.

John D. Cook, Jairo Fúquene, Luis Pericchi. Skeptical and optimistic robust priors for clinical trials, Revista Columbiana de Estadistica, (2011) 34 no. 2, pp. 333–345.

*Abstract*.
A useful technique from the subjective Bayesian viewpoint, suggested by Spiegelhalter et al. (1994), is to ask the subject matter researchers and other parties involved, such as pharmaceutical companies and regulatory bodies, for reasonable optimistic and pessimistic priors regarding the effectiveness of a new treatment. Up to now, the proposed skeptical
and optimistic priors have been limited to conjugate priors, though there is no need for this limitation. The same reasonably adversarial points of view can be taken with robust priors. A recent reference with robust priors usefully applied to clinical trials is in Fuquene, Cook, and Pericchi (2009). Our proposal in this paper is to use Cauchy and intrinsic robust priors for both skeptical and optimistic priors leading to results more closely related with the sampling data when prior and data are in conflict. In other words, the use of robust priors removes the dogmatism implicit in conjugate priors.

John D. Cook. Block Adaptive Randomization (2011). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 63.

*Abstract*. This note proposes a block-adaptive randomization method to limit the length of runs in an outcome-adaptive randomized trial.

John D. Cook Upper bounds on non-central chi-squared tails and truncated normal moments (2010). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 62.

*Abstract*.
We show that moments of the truncated normal distribution provide upper bounds on the tails of the non-central chi-squared distribution, then develop upper bounds for the former.

John D. Cook Asymptotic results for Normal-Cauchy model (2010). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 61.

*Abstract*. This report proves asymptotic results for the posterior mean when sampling from a normal distribution with a Cauchy prior on the location parameter.

John D. Cook How to test a random number generator (2009). Chapter 10 from Beautiful Testing: Leading Professionals Reveal How They Improve Software

John D. Cook Determining distribution parameters from quantiles

*Abstract*. Bayesian statistics often requires eliciting
prior probabilities from subject matter experts who are unfamiliar with statistics.
While most people an intuitive understanding of the mean of a probability distribution,
fewer people understand variance as well, particularly in the context of asymmetric
distributions. Prior beliefs may be more accurately captured by
asking experts for quantiles rather than for means and variances.
This note will explain how to solve for parameters so that common
distributions satisfy two quantile conditions. We present algorithms for
computing these parameters and point to corresponding software.
The distributions discussed are normal, log normal, Cauchy, Weibull,
gamma, and inverse gamma. The method given for the normal and
Cauchy distributions applies more generally to any location-scale family.

John D. Cook. Exact calculation of inequality probabilities (2009). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 54.

*Abstract*.
This note surveys results for computing the inequality probability P(X > Y) in closed form where X and Y are independent continuous random variables. Distribution families discussed include normal, Cauchy, gamma, inverse gamma, Levy, folded normal, and beta. Mixture distributions are also discussed.

Jairo A. Fúquene P., John D. Cook, Luis Raúl Pericchi. A Case for Robust Bayesian priors with Applications to Binary Clinical Trials. Bayesian Analysis (2009) 4, Number 4, pp. 817–846.

*Abstract*.
Bayesian analysis is frequently limited to conjugate Bayesian analysis,
particularly in the case in the analysis of clinical trial data.
Even though conjugate analysis may be simpler computationally,
the price to be paid is high: such analysis is not robust with respect to the prior,
*i.e.,* changing the prior may affect the conclusions without bound.
Furthermore conjugate Bayesian analysis is blind with respect to the potential conflict
between the prior and the data. On the other hand, robust priors have bounded influence.
The prior is discounted automatically when there are conflicts between prior information and data.
The original proposal of robust priors was made by de-Finetti in the 1960's.
However, the practice has not taken hold in important areas such as in
clinical trials where conjugate priors are ubiquitous.

We show here how the Bayesian analysis for simple binary binomial data, after expressing in its exponentially family form, is improved by employing Cauchy priors. Moreover, we also introduce in the analysis of clinical trials a robust prior originally developed by J.O. Berger that gives closed-form results when coupled with a normal log-odds likelihood. Berger's prior yields the superior robust analysis with no added computational complication compared to the conjugate analysis. We illustrate the results with famous textbook examples and a with data set and a prior from a previous trial.

On the formal side, we give here a theorem that we call the “Polynomial Tails Comparison Theorem.”
This theorem establishes the analytical behavior of any likelihood function with tails bounded by
a polynomial when used with priors with polynomial-order tails, such as Cauchy or Student's *t*.
The likelihood does not have to be a
location family nor exponential family distribution and the conditions of the theorem are easily verifiable.
For Berger's prior robustness can be established directly since the exact expressions for posterior moments are known.

John D. Cook. Inequality Probabilities for Folded Normal Random Variables (2009). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 52.

*Abstract*.
This note explains how to calculate the probability Pr(|X| > |Y|)
for normal random variables X and Y.
(A random variable formed by taking the absolute value of a normal random variable is known as a *folded* normal random variable.)
When X and Y have equal variance, a simple expression is obtained. Otherwise the problem is reduced to a well-known problem.

Valen E. Johnson, John D. Cook. Bayesian Design of Single-Arm Phase II Clinical Trials with Continuous Monitoring. Clinical Trials 2009; 6(3):217–26. (preprint)

*Abstract*.
Many “Bayesian” clinical trial designs use posterior credible intervals as
tools to define stopping boundaries for inferiority, futility, or superiority.
However, the thresholds on posterior credible intervals that trigger termination
of a trial are determined by frequentist operating characteristics. This
practice can result in substantial overlap between the credible intervals
associated with, say, stopping a trial for superiority and stopping a trial for
inferiority, which severely limits the interpretation of posterior probability
statements. In this article, we use formal Bayesian hypothesis tests to design
single-arm phase II clinical trials. By using non-local prior densities to
define null and alternative models, we obtain exponential convergence of Bayes
factors under both null and alternative models. When compared to other commonly
used Bayesian and frequentist designs, we show that our method provides better
operating characteristics, uses fewer patients per correct decision, and
provides more directly interpretable results. We also demonstrate that designs
based on Bayesian hypothesis tests eliminates a potential source of bias often
associated with Bayesian trial designs.

John D. Cook, Luis Raúl Pericchi. Information and Cross-Entropic Approaches, In: Lauretto MS; Pereira CAB; Stern JM (Org.), Bayesian Methods and Maximum Entropy Methods in Science and Engineering 28, Melville: AIP — American Institute of Physics, 2008, v 28, pp 278–285.

*Abstract*.
In a recent working paper, Fúquene, Cook, and Pericchi
make a comprehensive proposal putting forward robust, heavy-tailed
priors over conjugate, light-tailed priors in Bayesian analysis.
The paper focuses particularly on clinical trials, where
information from previous trials should be used in an non-dogmatic
fashion, suggesting the use of robust priors. Robust priors
have bounded influence in the posterior distribution and their
influence is inversely related to the conflict between the data in
previous and the current trial.
Clearly, the likelihood has to be taken into consideration and not
only the prior. We explore here a novel proposal based on a
cross-entropy measure of comparison between different models, on
which the expectations of the log ratio of the evidences of the
contender models are taken with respect to each of the different
models and then compared. We also compute the expected increase of
information within each model. Both criteria seems to justify the
use of robust priors.

John D. Cook.
Exact operating characteristics for single-arm Phase II trials (2008).
*UT MD
Anderson Cancer Center Department of Biostatistics Working Paper Series.*
Working Paper 45.

*Abstract*. Simulation is so widely used in studying
the operating characteristics of clinical trials that we may forget that
simulation is not always necessary. This note gives an algorithm for computing
the operating characteristics of a stopping rule for a single-arm phase II
clinical trial exactly.

Yisheng Li, Benjamin Bekele, Yuan Ji, and John D. Cook. Dose-Schedule Finding in Phase I/II Clinical Trials Using Bayesian Isotonic Transformation (2008). Statistics in Medicine 27:4895–4913

*Abstract*.
The intent of most phase I oncology trials is to determine
the maximum-tolerated dose (MTD) of an experimental treatment.
One of the main considerations apart from determining the MTD is
determining an appropriate schedule for administration of the
treatment. Historically, schedules have been fixed prior to the
start of dose finding. Recently, an increasing number of trials
have been designed to determine the MTDs during a phase I
component and subsequently determine a schedule during a phase
II component. In this paper, we propose a Bayesian design for
dose-schedule finding by jointly modeling binary toxicity and
efficacy outcomes. Assuming the probability of toxicity follows
an order constraint between schedules, we apply a Bayesian
isotonic transformation approach to estimating the constrained
parameters. We select a dose-schedule combination based on the
joint posterior distribution of toxicity and efficacy. Using
simulation studies for a hypothetical and a practical cancer
clinical trial, we demonstrate that the proposed design performs
well under different clinical scenarios.

John D. Cook. The
Effect of Population Drift on Adaptively Randomized Trials (2007). *UT MD
Anderson Cancer Center Department of Biostatistics Working Paper Series.*
Working Paper 39.

*Abstract*. Adaptively randomized trials aim to treat patients in clinical trials more
effectively by increasing the probability of assigning treatments that appear to
have a higher probability of response. Studies of adaptive randomization to date
have assumed constant probabilities of response on each treatment. This paper
examines the effect of response probabilities that change over time due to
population drift.

Harry T. Whelan et al,
Practical model-based dose-finding in early phase
clinical trials: Optimizing tPA dose for treatment of ischemic stroke in
children. *Stroke*.
39 (2008) 2627-2636.

*Abstract*. A safe and effective tissue plasminogen activator (tPA) dose for childhood
stroke has not been established. This paper describes a Bayesian
outcome-adaptive method for determining the best dose of an experimental agent,
and explains how this method was used to design a dose-finding trial for tPA in
childhood acute ischemic stroke (AIS).

J. Kyle Wathen, Peter F. Thall, John D. Cook, Elihu H. Estey, Accounting for Patient Heterogeneity in Phase II Clinical Trials. Statistics in Medicine. 27 (2008) 2802–2815.

*Abstract*. Phase II clinical trials typically are single-arm studies conducted to decide
whether an experimental treatment is sufficiently promising, relative to
standard treatment, to warrant further investigation. Many methods exist for
conducting phase II trials under the assumption that patients are homogeneous.
In the presence of patient heterogeneity, however, these designs are likely to
draw incorrect conclusions. We propose a class of model-based Bayesian designs
for single-arm phase II trials with a binary or time-to-event outcome and two or
more prognostic subgroups. The designs' early stopping rules are subgroup
specific and allow the possibility of terminating some subgroups while
continuing others, thus providing superior results when compared with designs
that ignore treatment-subgroup interactions. Because our formulation requires
informative priors on standard treatment parameters and subgroup main effects,
and non-informative priors on experimental treatment parameters and
treatment-subgroup interactions, we provide an algorithm for computing prior
hyperparameter values. A simulation study is presented and the method is
illustrated by a chemotherapy trial in acute leukemia.

Marcos de Lima et al, Phase I/II study of gemtuzumab ozogamicin added to fludarabine, melphalan and allogeneic hematopoietic stem cell transplantation for high-risk CD33 positive myeloid leukemias and myelodysplastic syndrome. Leukemia 22 (2008) pp 258–264.

*Abstract*. We investigated the hypothesis that
gemtuzumab ozogamicin (GO), an anti-CD33 immunotoxin would
improve the efficacy of fludarabine/melphalan as a
preparative regimen for allogeneic hematopoietic stem cell
transplantation (HSCT) in a phase I/II trial. Toxicity was
defined as grades III–IV organ damage, engraftment failure
or death within 30 days. 'Response' was engraftment and
remission (CR) on day +30. We sought to determine the GO
dose (2, 4 or 6 mg m^{-2}) giving the best trade-off
between toxicity and response. All patients were not
candidates for myeloablative regimens. Treatment plan: GO
(day -12), fludarabine 30 mg m^{-2} (days -5 to -2),
melphalan 140 mg m^{-2} (day -2) and HSCT (day 0).
GVHD prophylaxis was tacrolimus and mini-methotrexate.
Diagnoses were AML (*n*=47), MDS (*n*=4) or CML (*n*=1).
Median age was 53 years (range, 13–72). All but three
patients were not in CR. Donors were related (*n*=33)
or unrelated (*n*=19). Toxicity and response rates at
4 mg m^{-2} were 50% (*n*=4) and 50% (*n*=4).
GO dose was de-escalated to 2 mg m^{-2}: 18% had
toxicity (*n*=8) and 82% responded (*n*=36).
100-day TRM was 15%; one patient had reversible hepatic VOD.
Median follow-up was 37 months. Median event-free and
overall survival was 6 and 11 months. GO 2 mg m^{-2}
can be safely added to fludarabine/melphalan, and this
regimen merits further evaluation.

John D. Cook,
Comparing Methods of Tuning Adaptively Randomized Trials (2007). *UT MD
Anderson Cancer Center Department of Biostatistics Working Paper Series.*
Working Paper 32.

*Abstract*.
The simplest Bayesian adaptive randomization scheme is to randomize
patients to a treatment with probability equal to the probability p
that the treatment is better. We examine three variations on
adaptive randomization which are used to compromise between this
scheme and equal randomization. The first variation is to apply a
power transformation to p to obtain randomization probabilities. The
second is to clip p to live within specified lower and upper bounds.
The third is to begin the trial with a burn-in period of equal
randomization. We illustrate how each approach effects statistical
power and the number of patients assigned to each treatment. We
conclude with recommendations for designing adaptively randomized
clinical trials.

John D. Cook,
Understanding the Exponential Tuning Parameter in Adaptively Randomized Trials
(2006). *UT MD Anderson Cancer Center Department of Biostatistics Working
Paper Series.* Working Paper 27.

*Abstract*.
We examine the effect of a parameter λ used to
calibrate how responsive randomization probabilities are to observed data in an
adaptively randomized clinical trial. We define and motivate the parameter
λ and demonstrate how varying this parameter effects the operating
characteristics of example clinical trial designs.

John D. Cook and Saralees Nadarajah. Stochastic Inequality Probabilities for Adaptively Randomized Clinical Trials. Biometrical Journal. 48 (2006) pp 256–365.

*Abstract*. We examine stochastic inequality probabilities of the form P(X > Y) and P(X >
\max(Y, Z)) where X, Y, and Z are random variables with beta, gamma, or inverse
gamma distributions. We discuss the applications of such inequality
probabilities to adaptively randomized clinical trials as well as methods for
calculating their values.

Peter F. Thall and John D. Cook. Using both efficacy and toxicity for dose-finding. In S. Chevret (ed), Statistical Methods for Dose Finding Experiments. New York: John Wiley & Sons, June 2006.

Peter F. Thall and John D. Cook. Adaptive dose-finding based on efficacy-toxicity trade-offs. Encyclopedia of Biopharmaceutical Statistics, 2nd edition, 2006, Chein-Chung Chow editor.

Peter F. Thall, John D. Cook, and Elihu H. Estey. Adaptive dose selection using efficacy-toxicity trade-offs: illustrations and practical considerations. J Biopharmaceutical Stat. 16: 623–638 (2006)

*Abstract*.
The purpose of this paper is to describe and illustrate an outcome-adaptive Bayesian
procedure, proposed by Thall and Cook (2004), for assigning doses of an experimental
treatment to successive cohorts of patients. The method uses elicited (efficacy, toxicity)
probability pairs to construct a family of trade-off contours that are used to quantify
the desirability of each dose. This provides a basis for determining a best dose for each
cohort. The method combines the goals of conventional Phase I and Phase II trials,
and thus may be called a “Phase I-II” design. We first give a general review of the
probability model and dose-finding algorithm. We next describe an application to a trial
of a biologic agent for treatment of acute myelogenous leukemia, including a computer
simulation study to assess the design's average behavior. To illustrate how the method
may work in practice, we present a cohort-by-cohort example of a particular trial. We
close with a discussion of some practical issues that may arise during implementation.

John D. Cook.
Efficacy-toxicity trade-offs based on *L ^{p}* norms (2006).

*UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series.*Working Paper 29.

*Abstract*.
This report examines in detail a family of efficacy-toxicity
trade-off functions simpler and more general than those
originally proposed in [1]. The new trade-off functions are
based on distance in Lp norm to the ideal point and were first
presented in [2]. We define and illustrate these functions and
demonstrate how to compute their parameters based on elicited
values.

J. Kyle Wathen and John D. Cook. Power and bias in adaptively randomized clinical trials (2006). Technical Report UTMDABTR-002-06.

*Abstract*. This report examines the operating characteristics of adaptively
randomized trials relative to equally randomized trials in regard to power and
bias. We also examine the number of patients in the trial assigned to the
superior treatment. The effects of prior selection, sample size, and patient
prognostic factors are investigated for both binary and time-to-event outcomes.

John D. Cook.
Numerical evaluation of gamma inequalities (2006).
*UT MD Anderson Cancer Center Department of Biostatistics Working
Paper Series.* Working Paper 30.

*Abstract*. This paper addresses the problem of numerically evaluating the probabilities
P(X > Y), P(X > max(Y,Z)), and P(X < min(Y,Z)) where X, Y, and Z are independent
gamma or inverse gamma random variables.

John D. Cook. Continuous safety monitoring in single-arm, time-to-event trials without software (2005). Technical Report UTMDABTR-006-05.

*Abstract*. This note concerns trial conduct for one-arm trials that monitor
safety by comparing time-to-event outcomes of the experimental treatment to an
historical treatment. To date, such trials have been conducted using software
which evaluates the stopping rule as the trial progresses. We show that is it
possible to pre-calculate the stopping conditions, simplifying trial conduct and
opening up new possibilities.

John D. Cook. Exact calculation of beta inequalities (2005). Technical Report UTMDABTR-005-05.

*Abstract*. This paper addresses the problem of evaluating P(X > Y) where X and Y are
independent beta random variables. We cast the problem in terms of a
hypergeometric function and use hypergeometric identities to calculate the
probability in closed form for certain values of the distribution parameters.

Peter F. Thall and John D. Cook. Dose-finding based on efficacy-toxicity trade-offs (2004) Biometrics, 60:684–693.

*Abstract*. We present an adaptive Bayesian method for dose-finding in phase I/II
clinical trials based on trade-offs between the probabilities of treatment
efficacy and toxicity. The method accommodates either trinary or bivariate
binary outcomes, as well as efficacy probabilities that are potentially non-monotone in
dose. Doses are selected for successive patient cohorts based on a set of
efficacy-toxicity trade-off contours that partition the two-dimensional outcome
probability domain. Priors are established by solving for hyperparameters which
optimize the fit of the model to elicited mean outcome probabilities. For trinary
outcomes, the new algorithm is compared to the method of Thall and Russell by
application to a trial of rapid treatment for ischemic stroke. The bivariate
binary outcome case is illustrated by a trial of graft-versus-host disease
prophylaxis in allogeneic bone marrow transplantation. Computer simulations show
that, under a wide rage of dose-outcome scenarios, the new method has high
probabilities of making correct decisions and treats most patients at doses with
desirable efficacy-toxicity trade-offs.

John D. Cook. Simulation results for phase II clinical trial durations (2004). Technical Report UTMDABTR-014-04

*Abstract*. This paper investigates the effect of cohort size on phase II clinical trial
duration by doing a simulation study of a monitoring method of Thall and Simon.
We challenge the assumptions that larger cohort sizes lead to shorter trials and
that continuous monitoring is impractical.

John D. Cook.
Numerical
computation of stochastic inequality probabilities (2003).
*UT MD Anderson Cancer Center Department of Biostatistics Working
Paper Series.* Working Paper 46.

*Abstract*. This paper addresses the problem of numerically evaluating P(X > Y) for
independent continuous random variables X and Y. This calculation arises in the
design of clinical trials and as such appears in the inner loop of simulations
of these trials. An early example of this is given in (Thompson 1933). More
recent examples are given in (Giles et al 2003), (Berry 2003a, 2003b). It is
worthwhile to optimize the calculation of these probabilities as they may be
computed millions of times in the course of simulating a single trial.
Techniques such as memoization (Orwant 2002) can eliminate redundant
calculations of such probabilities over a simulation but the need for a large
number of evaluations remains. After considering how to compute P(X > Y) in
general, we present optimizations for
important special cases in which X and Y both belong to one of the following
families of classical distributions: exponential, gamma, inverse gamma, normal,
Cauchy, beta, and Weibull.

John D. Cook and Ralph E. Showalter. Microstructure Diffusion Models with Secondary Flux (1995) Journal of Mathematical Analysis 1995, pp 731–756

*Abstract*. Totally fissured media in which the cells are isolated by the fissure system
are effectively described by double porosity models with microstructure. These
models contain the geometry of the individual cells or pores in the medium and
the flux across their interface with the fissures which surround them. We extend
these models to include the case of partially fissured media in which a
secondary flux effect arises from cell-to-cell diffusion paths. These
quasi-linear problems are formulated in appropriate spaced for which the cells
respond to the local linearization of the fissure pressure. It is shown that
they are well-posed and the solutions depend continuously on parameters that
determine the models.

John D. Cook and Ralph E. Showalter. Distributed Systems of PDE in Hilbert Space (1993) Differential and Integral Equations, Vol 6 No 5, Sept 1993, pp. 981–994

*Abstract*. We present a system of two nonlinear evolution equations and
a corresponding approximating system which provide a common framework for
studying *distributed microstructure models* and a variety of other models
for transport and diffusion in heterogeneous media. Existence and uniqueness are
demonstrated using semigroup methods, and solutions to the approximating system
are shown to converge strongly to the solution of the limiting system. In the
microstructure case, new results are obtained, and additional PDE examples are
provided to show that in general, certain hypotheses cannot be removed.

John D. Cook. A Stefan Problem on a Region and its Boundary (1993) Applicable Analysis, Vol 57 No 3–4 (95) pp 367–381

*Abstract*. This paper considers a system of equations consisting of a
nonlinear evolution equation on an open set Ω in R^{n} coupled to another
nonlinear evolution equation on the boundary ∂Ω.
Rather general assumptions are made concerning the operators involved and the coupling between the two problems.
Existence and uniqueness are demonstrated via a semigroup of nonlinear operators
on L^{1}(Ω) x L^{1}(∂ Ω).

John D. Cook. Separation of convex sets in linear topological spaces (1988)

*Abstract*. This paper discusses under what conditions two disjoint convex subsets of a
linear topological space can be separated by a continuous linear functional. The
equivalence of several forms of the Hahn-Banach theorem is proven. The
separation problem is considered in linear topological spaces, locally convex
linear topological spaces, Banach spaces, and finally finite dimensional Banach
spaces. A number of examples are included to show the necessity of the
hypotheses of various theorems.