Kurt L Helmes
email: helmes@wiwi.hu-berlin.de

Richard H Stockbridge
email: stockbri@csd.uwm.edu

Thinning and harvesting in stochastic forest models

Keywords: stochastic forest models, forest rotation, Wicksell, Faustmann, harvest, thinning, linear programming JEL: C61, Q23

This paper analyzes a stochastic forest growth model in which the manager is able to first thin the forest to promote better growth before harvesting. Both Wicksell single thinning-and-harvesting cycle and Faustmann on-going rotation problems are considered. The Wicksell problem is analyzed by first restricting the class of decision times to (thinning,harvesting) pairs that bound the growth away from infinity and imbedding the problem in an infinite-dimensional linear program on a space of triplets of measures. These measures capture the thinning and harvesting decisions along with the behavior of the growth process prior to harvest. An auxiliary linear program then leads to a nonlinear optimization problem for which an optimal value and solution are determined. The values of all the problems are be related through a set of inequalities. The solution of the nonlinear problem determines (random) thinning and harvesting times for the single thinningand-harvesting cycle which demonstrate the equality of the values of these various problems. Finally for the Wicksell problem, the unrestricted class of thinning-and-harvest times is shown to give the same value as the restricted class. The Faustmann on-going thinning-and-harvesting rotation problem is reduced to a Wicksell problem which then allows for the characterization of the value as the solution to a different nonlinear optimization problem. The effects of the opportunity to thin the forest are illustrated on a mean-reverting stochastic model.

Introduction

Consider a stochastic forest model where, without any intervention, the process X is assumed to satisfy dX(t) = μ(X(t), Y (t)) dt + σ(X(t), Y (t)) dW (t), X(0) = x new > 0.

(1)

Here X is a process that captures the volume and quality of the forest stand, x new is the value of a "new" forest, μ denotes the mean growth rate, σ denotes the volatility of the growth rate and W is a standard Brownian motion process which provides the random fluctuations of the forest size. The process Y is an indicator process that identifies whether the forest is new and densely planted (Y (t) = 1) or has been thinned (Y (t) = 2); we assume Y (0) = 1. For our model, we have in mind functions μ and σ such that the process X is always nonnegative and represents a growth process. Allowing dependence on the process Y in these coefficients means that the growth dynamics can differ for dense and thinned forests or even that the process X can represent different quantities for the two types of forest. For example, X may represent the volume of fuel wood for a dense forest and the diameter at breast height (dbh) for an average, more valuable tree in the thinned forest. The interventions in the growth of the forest occur when it is thinned and harvested. This paper analyzes both a single thinning-harvesting cycle of Wicksell type and the Faustmann on-going rotation problem for this two-decision model.

For the Wicksell single-cycle problem, the goal of the forest manager is to maximize the expected presentvalue of the net proceeds of the forest product when thinned and harvested. Let θ denote the time at which the manager thins the forest and let η be the harvest time. The size of the forest after thinning is X(θ) whereas the size of the stand is X(θ-) at the time the decision to thin is made; in this model, the thinning decision results in an immediate jump to the process X. Let g 1 and g 2 denote the net profit functions from thinning and harvesting, respectively, and let α > 0 denote the discount rate. The single-cycle objective is one of selecting times θ and η, with θ ≤ η, so as to maximize E e -αθ g 1 (X(θ-))I {θ<∞} + e -αη g 2 (X(η))I {η<∞} .

(2)

Let V w (x new , 1) denote the optimal value. At this point, one must mention the importance of the interpretation of the model for the dense and thinned states. The process X may possibly represent two different quantities in the two states, in which case it may be reasonable for thinning to result in an increase to the process X; X(θ) > X(θ-). When the model is such that X represents the same quantity for both dense and thinned forests, it may be necessary to impose an additional condition on the decision times of the model. For example, if X were to represent the volume of lumber on the stand, then a decision to thin would reduce the size of X and the model would therefore restrict the thinning decision to occur only after the process X exceeds the size of the thinned stand. If X(θ) = x thin , in the simplest case, then one would require θ > τ xthin , where τ xthin is the first time at which X achieves level x thin . This paper treats the more general model by allowing X to increase in value when thinning occurs; the results can be easily adapted to include additional restrictions on the decision times.

The Faustmann rotation problem replants trees for the cycle to repeat with the new planting resulting in the forest stand returning to value x new so the process X starts from this point following a harvest decision. For k = 1, 2, 3, . . ., let θ k and η k denote the random times at which the forest is thinned and harvested, respectively, for the k th time. Then for each k, X(θ k -) and X(η k -) represent the values of X when the decisions are made to thin and harvest, respectively, while X(θ k ) and X(η k ) = x new denote the sizes of X immediately following these interventions. The Faustmann on-going rotation problem is one of maximizing E ∞ k=1 e -αθk g 1 (X(θ k -))I {θk<∞} + e -αηk g 2 (X(η k -))I {ηk<∞} .

(3) Let V f (x new , 1) denote the optimal value for the Faustmann problem.

The mathematical modeling of forest growth with the aim of determining on-going optimal harvesting decisions began with the paper by [START_REF] Faustmann | Berechnung des Werthes, welchen Waldboden, sowie noch nicht haubare Holzbestände für die Waldwirthschaft besitzen[END_REF] which considered the case of deterministic growth. Since 1849, a large literature has developed on this topic. The bibliography by [START_REF] Newman | Forestry's golden rule and the development of the optimal forest rotation literature[END_REF] provides a partial but extensive up-to-date (2002) list of references on the economics of forest rotation. We concentrate our comments on some papers using stochastic models. [START_REF] Nordstrøm | A stochastic model for the growth period decision in forestry[END_REF] modelled the growth process deterministically but introduced randomness with prices that followed a finite-state Markov chain in discrete time. [START_REF] Miller | A stochastic analysis of the tree paradigm[END_REF] consider a diffusion process for the tree size and solve the rotation problem. A limitation of this model is that tree sizes will become negative. [START_REF] Clarke | The tree-cutting problem in a stochastic environment[END_REF] use a geometric Brownian motion for the forest size to ensure positivity and an age-dependent geometric Brownian motion for the price process. Using optimal stopping methods, the paper analyzes both the Wicksell single-period and Faustmann on-going harvest rotation problems. [START_REF] Willassen | The stochastic rotation problem: A generalization of Faustmann's formula to stochastic forest growth[END_REF] considers a general stochastic differential equation model for the growth process in continuous time and uses impulse control methods to solve the problem. [START_REF] Buongiorno | Generalization of Faustmann's formula for stochatic forest growth and prices with Markov decision process models[END_REF] and others employ Markov decision processes to model the growth process in discrete time; Buongiorno reformulates the problem as a finite-dimensional linear program. [START_REF] Sødal | The stochastic rotation problem: A comment[END_REF] restricts his analysis to decision times that are hitting times of the growth process at fixed levels and uses an intuitive mark-up pricing approach to characterize the value as a nonlinear optimization problem. Additional work on the Wicksell single-period problem for stochastic growth models include papers by Alvarez and Koskela (see, e.g., Alvarez and Koskela (2005)), among others. [START_REF] Penttinen | Impact of stochastic price and growth processes on optimal rotation age[END_REF] includes thinning considerations in his models but only in terms of cost, not with the possibility of the growth dynamics improving.

Though our exposition is expressed entirely in terms of forests, their harvest, rotation and thinning, the paper by [START_REF] Miller | A stochastic analysis of the tree paradigm[END_REF] discusses how the rotation problem is a paradigm that has many additional economic applications.

The thinning and harvest decisions represent times when the decision maker elects to switch from one diffusion process to the other. Switching problems have also been considered in the literature where from a mathematical point of view such decisions are phrased in terms of optimal stopping, impulse control or more general stochastic control problems. The traditional approach is one of solving the Hamilton-Jacobi-Bellman (HJB) equation or variational or quasi-variational inequalities associated with the given problem (see, e.g., [START_REF] Baldursson | Irreversible investment and industry equilibrium[END_REF], [START_REF] Brekke | Optimal Switching in an Economic Activity under Uncertainty[END_REF], [START_REF] Duckworth | A Model for Investment Decisions with Switching Costs[END_REF], [START_REF] Lumley | A Model for Investments in the Natural Resource Industry with Switching Costs[END_REF], [START_REF] Zervos | A Problem of Sequential Entry and Exit Problems Combined with Optimal Stopping[END_REF] and references therein, for a short selection of recent publications). This indirect approach seeks the solutions to a family of optimization problems that are parametrized by the initial position of the process. The mathematical challenge is to prove optimality within a large class of policies and to prove the existence of a solution of an HJB-equation or variational inequality under weak conditions and mild regularity assumptions imposed on the data.

The thinning-and-harvest problem is also similar to entry-and-exit problems in economics in which the decision maker determines times at which to enter a market and times at which to exit from the market. This type of problem was introduced by [START_REF] Brennan | Evaluating Natural Resource Investments[END_REF] and further analyzed by [START_REF] Dixit | Entry and Exit Decisions under Uncertainty[END_REF] and [START_REF] Dixit | Investment under Uncertainty[END_REF]. These papers, as do some of the ones mentioned in the previous paragraph, only consider a single diffusion process rather than switching diffusions.

This paper approaches the optimal switching problem quite differently by concentrating on the optimization problem corresponding to a single initial value rather than addressing the parametrized family of problems. The stochasticity of the problem is captured by occupation measures that must satisfy an identity (see (7) below) for a large class of test functions. The class is rich enough that one is able to reconstruct the stochastic process, though this paper does not take advantage of this fact; see, for example, (Kurtz andStockbridge, 1998, 2001) for more details. The measures become the variables for an infinite-dimensional linear program that provides an upper bound on the optimal value. The constraints of this linear program are then relaxed to obtain an auxiliary linear program for which a further upper bound is easily determined. This final upper bound arises as the value of a nonlinear optimization problem in two variables. Using optimal values of these variables allows one to determine sequences of decision times such that the value of the original problem equals the highest upper bound and hence these times are seen to be optimal.

This paper is organized as follows. Section 2 concentrates on the Wicksell single thinning-and-harvesting cycle problem, beginning by placing a restriction on the class of decision times. The restricted stochastic problem is imbedded in an infinite-dimensional linear program in Section 2.1 having variables in a space of triplets of measures. Section 2.2 develops the auxiliary linear program and corresponding nonlinear optimization problem in such a manner that all of the values for the various problems can be related by inequalities. It also characterizes an optimal solution within the class of restricted decision times. The unrestricted problem is shown to have the same value in Section 2.3 and hence the optimal thinning and harvest times for the restricted problem are seen to be optimal for the unrestricted problem as well. Section 3 examines the Faustmann thinning-and-harvesting rotation problem. The strong Markov property is employed in Section 3.1 to reduce the Faustmann problem to a Wicksell single-cycle problem with a slight modification to the harvest payoff function. As a result, the value is characterized as the solution to a different nonlinear optimization problem. Illustrative examples using a mean-reverting growth process are given in Sections 2.4 and 3.2.

The imbedding of the stochastic problem in an infinite-dimensional linear program can often be sharpened to show equivalence between the formulations. [START_REF] Kurtz | Existence of Markov controls and characterization of optimal Markov controls[END_REF] establishes this equivalence for absolutely continuous stochastic control problems, [START_REF] Cho | Linear programming formulation for optimal stopping problems[END_REF] proves this for optimal stopping problems in which the processes exclude singular behavior (such as the thinning decisions of this paper) and [START_REF] Helmes | Linear Programming Approach to the Optimal Stopping of Singular Stochastic Processes[END_REF] extends this result to processes having singular behavior. These results are proven with less regularity of the payoff functions, typically only requiring semi-continuity. The linear programming formulation then allows one to employ numerical techniques to approximate the optimal solutions. A variety of such numerical approaches are possible, including approximating the diffusion process by a continuous-time Markov chain, characterizing the measures using their moments and using finiteelements to determine densities for approximating measures. The determination of the occupation measures of an optimal process can then be used to determine other quantities of interest about the process. The model and analysis of this paper do not require the equivalence since the equality of the values is derived from the exact analysis.

The linear programming approach extends, for example, to the type of problem considered by [START_REF] Lumley | A Model for Investments in the Natural Resource Industry with Switching Costs[END_REF] in which in addition to the times at which to begin production or shut it down, the decision maker needs to select the level of production as well. For such problems, the occupation measures would include a component related to the level of production.

Detailed Formulation

We begin with a precise formulation of the growth model with thinning. Assume the coefficients μ and σ are continuous and are such that the process X y , y = 1, 2, is a weak solution of the stochastic differential equation

dX y (t) = μ(X y (t), y) dt + σ(X y (t), y) dW (t), X y (0) = 0, (4)
(see Ethier and Kurtz, 1986, Section 5.3, pg. 291, for details) and that these solutions are unique in distribution. Let A y denote the generator of the process X y given by

A y f (x) = μ(x, y)f (x) + (σ 2 (x, y)/2)f (x).
The uniqueness in distribution then implies that the martingale problems for A y are well-posed and hence that each X y is a strong Markov process (see Ethier and Kurtz, 1986, Theorem 4.4.2, pg. 184). We wish to take advantage of the strong Markov property to piece together the solutions X 1 and X 2 to form a weak solution (X, Y ) of ( 1) at all times other than thinning and harvest times. Let {F t } denote a common filtration with respect to which X 1 anf X 2 are weak solutions of (4) for their respective values of y.

For the Wicksell problem, recall θ and η denote the thinning and harvesting times, respectively; θ and η are required to be {F t }-stopping times with θ ≤ η. For a given pair (θ, η), define the paired process (X, Y ) as follows. The initial values are X(0) = x new and Y (0) = 1. For 0 < t < θ, X(t) = x new + X 1 (t) and Y (t) = 1. At time θ, X jumps from X(θ-) to X(θ), where X(θ) has distribution π on a bounded interval [x min , x max ], with x min > 0, and Y (θ) = 2. The thinned level of the forest X(θ) is assumed to be independent of the thinning time θ. For t > θ, X(t) = X(θ) + X 2 (tθ). Since the Wicksell payoff ends at the harvest time, we may stop the process at η and specify X(η) = X(η-).

The Faustmann model is quite similar, but with a couple of important changes. The thinning and harvest times are {θ k } and {η k }, respectively. Define η 0 = 0. These decision times must be {F t }-stopping times that satisfy

η k-1 ≤ θ k ≤ η k , for each k ∈ N. Again, for each k ∈ N, let {X (k)
y } be a sequence of independent processes satisfying (4), y = 1, 2. At each harvest time

η k-1 , set (X(η k-1 ), Y (η k-1 )) = (x new , 1) and for η k-1 ≤ t < θ k , set X(t) = x new + X (k) 1 (t -η k-1 ) and Y (t) = 1. At each thinning time θ k , X(θ k ) is chosen from [x min , x max ] according to π, independently of {θ k } and {η k } and Y (θ k ) = 2. Then for θ k ≤ t < η k , define X(t) = X(θ k ) + X (k) 2 (t -θ k ) and Y (t) = 2.
Notice that at the times θ k , the forest is thinned to a level X(θ k ) within some range and this is assumed to happen instantaneously, and at the times η k , the forest is instantaneously harvested and replanted so X(η k ) = x new , with a corresponding decrease X(η k ) -X(η k -) to the process. Observe for each cycle, the process has "poorer growth" dynamics (4) with y = 1 between η k-1 and θ k and "better growth" dynamics (4) with y = 2 for times between θ k and η k .

Let A denote the set of pairs of admissible decision rules (θ, η) for the Wicksell problem and, with a slight abuse of notation, let A also denote the sequence of pairs {(θ k , η k )} for the Faustmann problem.

We place additional restrictions on the coefficients μ and σ through the behavior of the process at the boundaries. Assume ∞ is a natural boundary point (see (Borodin and Salminen, 2002, II.1.6, pp. 14, 15) or (Itô and McKean, Jr., 1974, pp. 128-131)) so that the forest does not grow without bounds in a finite time. Also assume that 0 is either a natural, entrance-not-exit or exit-not-entrance boundary point. In the former two cases, the forest will never die out in finite time (since x new > 0) whereas the last condition implies that once the forest fails, it never recovers on its own. For models in which 0 is an exit boundary point, the objective function would be non-positive for those decision times θ and η that exceed the hitting time ζ of X at 0.

We note that {F t } is the filtration associated with the weak solution to (1) so it may contain more information than that arising from the observations of the process X. Since the stopping times in A are {F t }-stopping times, these may in principle be determined using information contained in {F t } that is not generated by X. Our results nevertheless show that optimal decision rules exist within the subclass of hitting times of the process.

Before addressing the reward structure for the class of problems under consideration, it is important to describe some additional structure to the problem. The change in dynamics occurs only when the thinning and harvesting decisions are made and hence Y changes only at these decision times. As a result, except at the thinning and harvest decision times, the generator of the pair (X, Y ) is

Af (x, y) = μ(x, y) ∂f ∂x (x, y) + σ 2 (x,y) 2 ∂ 2 f ∂x 2 (x, y)
operating on functions f : R + × {1, 2} → R that are twice-continuously differentiable in x for each y. Due to the discounting at rate α, the eigenvalue problem

Af = αf (5)
plays a central role in the analysis. Under the conditions assumed in this paper, for each y ∈ {1, 2}, Af (•, y) = αf (•, y) has a nonnegative, strictly increasing solution ψ y (unique up to a positive multiplicative constant) and furthermore ψ y (0+) ≥ 0 and lim x→∞ ψ y (x) = ∞ (see Borodin and Salminen, 2002, II.1.10, pp. 18, 19). The specification of the model and the strictly increasing functions ψ 1 and ψ 2 are used to determine restrictions on the payoff functions g 1 and g 2 allowed in this paper.

Condition 1. The payoff functions g 1 and g 2 are assumed to satisfy the following:

(a) g 1 and g 2 are continuous and non-decreasing on [0, ∞);

(b) g 1 (x new ) < 0 and g 2 (x max ) ≤ 0;

(c) there exists some x < ∞ such that g 2 (x) > 0 and g 1 (x) + ψ2(xmin) ψ2(x) • g 2 (x) > 0; and

(d) for y = 1, 2, lim x→∞ g y (x) ψ y (x) = 0.
Condition 1(a) implies that a higher value of X yields a larger profit, whereas Condition 1(b) indicates there is non-zero cost to immediately thin a replanted forest and no profit from immediately harvesting the thinned forest. Condition 1(c) means that a sufficiently large value of X will return a positive profit for both harvesting and for thinning followed by harvesting; should this not be the case, then the optimal times θ and η would be infinite so that no cost is incurred. Finally, Condition 1(d) places a restriction on how quickly the reward rate can grow relative to the size of X. This condition will be needed in order to eliminate decisions to thin or harvest at arbitrarily large values from being near-optimal decisions.

Wicksell Single Thinning-and-Harvesting Cycle

This section analyzes the single-cycle problem of deciding when to thin and when to harvest a forest so as to maximize (2) over (θ, η) ∈ A with θ ≤ η. The examination of this problem, however, begins by restricting the decisions to a smaller collection A 1 of {F t }-stopping times. The restricted problem is imbedded in an infinite-dimensional linear program from which an auxiliary linear program is derived and a finite-dimensional nonlinear optimization problem arises. An optimal pair (θ * , η * ) of decision times is determined using an optimal solution to the nonlinear problem. Finally, the optimal solution for the restricted problem is shown to be optimal for the unrestricted problem and these results are illustrated by an example.

Let A 1 denote the collection of pairs of {F t }-stopping times (θ, η), with θ ≤ η, for which there exists some K < ∞ such that P (X(t) ≤ K, 0 ≤ t ≤ η) = 1. For such stopping times, the process X is bounded away from ∞. Note that K may differ for different (θ, η) ∈ A 1 . Let V w r (x new , 1) denote the optimal value over the restricted collection of pairs of stopping times.

Linear Program Imbedding

Select (θ, η) ∈ A 1 arbitrarily and define the process λ θ by λ θ (t) = I [0,t] (θ). Notice that λ θ starts at 0 and jumps to 1 at the random time θ after which it remains at 1.

Let D = C 2 c (R + × {1, 2}
), the space of twice-continuously differentiable functions having compact support. For f ∈ D, an application of Itô's formula results in

e -αt f (X(t), Y (t)) = f (x new , 1) + t 0 e -αs [Af -αf ](X(s), Y (s)) ds + t 0 e -αs [f (x, 2) -f (X(s-), Y (s-))] π(dx) dλ θ (s) + t 0 e -αs σ(X(s), Y (s)) ∂f ∂x (X(s), Y (s)) dW (s).
The conditions on f imply that the stochastic integral is a martingale so the optional sampling theorem (see Ethier and Kurtz, 1986, Theorem 2.2.13) establishes that

e -α(t∧η) f (X(t ∧ η), Y (t ∧ η)) -f (x new , 1) - t∧η 0 e -αs [Af -αf ](X(s), Y (s)) ds - t∧η 0 e -αs [f (x, 2) -f (X(s-), Y (s-))] π(dx) dλ θ (s) = f (x new , 1) + t∧η 0 e -αs σ(X(s), Y (s)) ∂f ∂x (X(s), Y (s)) dW (s)
and hence taking expectations and letting t → ∞ results in Dynkin's formula:

E e -αη f (X(η), Y (η))I {η<∞} -E η 0 e -αs [Af -αf ](X(s), Y (s)) ds -E η 0 e -αs [f (x, 2) -f (X(s-), Y (s-))] π(dx) dλ θ (s) (6) = f (x new , 1).
Now define three measures as follows: let ν η denote the discounted distribution of (X(η), Y (η)); let ν θ denote the discounted distribution of (X(θ-), Y (θ-)); and define the expected discounted occupation measure ν 0 so that for each

G 1 ∈ B(R + ) and y ∈ {1, 2} ν 0 (G 1 × {y}) = E η 0 e -αs I G1×{y} (X(s), Y (s)) ds .
Observe that Y (η) = 2 and Y (θ-) = 1 so ν η and ν θ can be (and, in the sequel, are) measures on R + . Notice also that the total masses of ν η and ν θ are bounded above by 1 and the total mass of ν 0 is bounded above by 1/α. Finally observe that the discounting implies that the sets {θ = ∞} and {η = ∞} contribute no mass to ν θ and ν η .

For a function h on [x min , x max ], define h, π = h(x) π(dx). Using the definitions of the measures ν θ , ν η and ν 0 along with the fact that π is a probability measure, (6) can be rewritten as

f (x, 2) ν η (dx) - [Af -αf ](x, y) ν 0 (dx × dy) - [ f (•, 2), π -f (x, 1)] ν θ (dx) = f (x new , 1) (7)
and this identity holds for all f ∈ D and (θ, η) ∈ A 1 . (The identity (7) also holds for all (θ, η) ∈ A but the ensuing argument requires the stopping times to be in A 1 .) These measures can be used to evaluate the expected payoff (2) resulting in

g 1 (x) ν θ (dx) + g 2 (x) ν η (dx). ( 8
)
Since for stopping times (θ, η) ∈ A 1 , the measures ν η , ν θ and ν 0 satisfy (7) and the corresponding value of ( 2) is given by ( 8), the stochastic decision problem on when to thin and harvest over the restricted class of decision rules is imbedded in the linear program

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Maximize g 1 dν θ + g 2 dν η Subject to f (x, 2) ν η (dx) -[Af -αf ](x, y) ν 0 (dx × dy) -[ f (•, 2), π -f (x, 1)] ν θ (dx) = f (x new , 1), ∀f ∈ D, 1 dν θ ≤ 1, 1 dν η ≤ 1, 1 dν 0 ≤ 1/α, ν θ , ν η , ν 0 measures. (9) Let V w lp (x new , 1
) denote the value of (9). The above argument immediately implies the following comparison of values.

Theorem 1. V w r (x new , 1) ≤ V w lp (x new , 1).

Auxiliary Linear Program and Nonlinear Optimization

The goal now is to simplify the linear program into an auxiliary linear program in a manner that becomes more tractable and for which the values can be easily related. Choose any (θ, η) ∈ A 1 and let K denote the bound corresponding to this pair. Recall, for each y ∈ {1, 2}, ψ y is a strictly increasing solution to Af (•, y) = αf (•, y). Let f (x, y) = a 1 ψ 1 (x)I {1} (y) + a 2 ψ 2 (x)I {2} (y) for some a 1 , a 2 ∈ R. This function does not have compact support so cannot be immediately used in (7). Let ξ : R + → R + be a mollifying function satisfying

ξ(x) = x for x ≤ K and ξ ∈ C 2 c (R + ). Considering the function f (x, y) = ξ(x) • f (x, y), we see that f ∈ D so can be used in (7). Moreover, for x ≤ K, ∂ f ∂x = ∂f ∂x and ∂ 2 f ∂x 2 = ∂ 2 f ∂x 2 . Since X(t) ≤ K a.s. for t ≤ η, it immediately follows that [A f -α f ](X(t), Y (t)) = 0 for all t ≤ η and therefore a 2 ψ 2 (x) ν η (dx) -[a 2 ψ 2 , π -a 1 ψ 1 (x)] ν θ (dx) = a 1 ψ 1 (x new ). ( 10
)
Thus when restricting the pairs of decision times (θ, η) to the subcollection A 1 , the identity (7) extends to the function f = a 1 ψ 1 I {1} + a 2 ψ s I {2} , for any a 1 , a 2 ∈ R.

We now define an auxiliary linear program by replacing the infinite collection of constraints in (9) by two constraints derived from (10) in which the first constraint selects a 1 = a 2 = 1, while the second constraint takes a 1 = 0 and a 2 = 1. Further relax the conditions by removing the total mass constraints on ν θ , ν η and ν 0 , thus eliminating ν 0 from the linear program. The auxiliary linear program is

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Maximize g 1 (x) ν θ (dx) + g 2 (x) ν η (dx) Subject to [ψ 1 (x) -ψ 2 , π ] ν θ (dx) + ψ 2 (x) ν η (dx) = ψ 1 (x new ), ψ 2 , π ν θ (dx) -ψ 2 (x) ν η (dx) = 0, ν η , ν θ measures. (11)
Inherent in the formulation ( 11) is the feasibility requirement on the measures that the function ψ 1 is integrable with respect to ν θ and ψ 2 is integrable with respect to ν η . Denoting the value of the auxiliary linear program by V w aux (x new , 1), again the next theorem follows from the above discussion. Theorem 2. V w lp (x new , 1) ≤ V w aux (x new , 1). We now turn to the analysis of (11) which is a special case of a general linear programming problem (A.1) in Appendix A. A remark is necessary, however, in order that Proposition 1 can be applied. One assumption on the model (A.1) is that the integrands in the constraints be positive. The only integrand in (11) for which this is questionable is ψ 1 (x)ψ 2 , π . Recall, the solutions ψ 1 and ψ 2 are unique up to a multiplicative constant so one may multiply ψ 1 by a sufficiently large constant in order to achieve this positivity; we assume the solution ψ 1 satisfies ψ 1 (x min ) > ψ 2 , π . Note this condition is satisfied when

ψ 1 (x min ) > ψ 2 (x max ). Theorem 3. Assume g 1 and g 2 satisfy Condition 1. Let F = {(u, v) : u ≥ x new , v ≥ x max }.
Then the value of the auxiliary linear program has the following bound:

V w aux (x new , 1) ≤ ψ 1 (x new ) • sup u,v∈F g 1 (u)ψ 2 (v) + ψ 2 , π g 2 (v) ψ 1 (u)ψ 2 (v) . ( 12
)
Moreover, optimizers (u * w , v * w ) of the right-hand side of ( 12) exist in F and the hitting times

θ * = inf{t > 0 : X(t-) = u * w } and η * = inf{t ≥ θ * : X(t-) = v * w } (13)
are optimal thinning and harvesting times, respectively, for the restricted Wicksell single-cycle problem. The optimal value for the restricted problem is

V w r (x new , 1) = g 1 (u * w )ψ 2 (v * w ) + ψ 2 , π g 2 (v * w ) ψ 1 (u * w )ψ 2 (v * w ) • ψ 1 (x new ). ( 14
)
Proof. Begin by normalizing the first constraint of (11) by dividing both sides by ψ 1 (x new ). The resulting formulation then matches the general linear program (A.1) so Proposition 1 can be applied, establishing the bound (12). Now rewrite the bounding ratio in (12) as

ψ 1 (x new ) ψ 1 (u) • g 1 (u) + ψ 2 , π ψ 2 (v) • g 2 (v)
from which one can see that optimizing over v only involves optimizing the ratio g2 (v) ψ2(v) . By Condition 1(c), there exists some x for which g 2 (x) > 0. The function ψ 2 is strictly positive on (0, ∞) so the ratio g2(x) ψ2(x) > 0. By continuity of g 2 and ψ 2 and Condition 1(d), the maximum value is achieved at some location v * w < ∞.

Now turning to the optimization over u, we seek to maximize the ratio

g1(u)+ ψ 2 ,π g 2 (v * w ) ψ 2 (v * w ) ψ1(u)
. Condition 1(c) implies this ratio will be strictly positive for some sufficiently large value of u. Since lim u→∞ ψ 1 (u) = ∞, the increase of g 1 in the numerator by the constant

ψ2,π g2(v * w ) ψ2(v * w )
does not affect the limiting value as u → ∞ and hence Condition 1(d) establishes the existence of a maximizer u * w < ∞. Finally, using a pair (u * w , v * w ) of maximizers of the ratio in ( 12), define the decision times θ * and η * as in (13). The measure ν θ * will concentrate all of its mass on {u * w } and similarly ν η * is a point mass on {v * w }. The two constraints of the auxiliary linear program (11) form the system

[ψ 1 (u * w ) -ψ 2 , π ] ν θ * ({u * w }) + ψ 2 (v * w ) ν η * ({v * w }) = ψ 1 (x new ), ψ 2 , π ν θ * ({u * w }) -ψ 2 (v * w ) ν η * ({v * w }) = 0 from which one readily determines that ν θ * ({u * w }) = ψ1(xnew) ψ1(u * w ) and ν η * ({v * w }) = ψ1(xnew) ψ1(u * w ) • ψ2,π ψ2(v * w )
. Hence this choice of thinning and harvesting times achieves the upper bound ( 14).

Remark 1. Let θ u = inf{t > 0 : (X(t-), Y (t-)) = (u, 1)} denote the first hitting time of u by X under dynamics given by Y (t) = 1 when X( 0

) = x new . It is well-known that E e -αθu = ψ 1 (x new ) ψ 1 (u) (15)
(see Borodin and Salminen, 2002, II.1.10, p. 18). Now let η v = inf{t > 0 : (X(t), Y (t)) = (v, 2)} be the first hitting time by X of v under dynamics given by Y (t) = 2. Conditioning on X(θ u ) = x, we have E e -α(ηv-θu) |X(θ u ) = x = ψ 2 (x)/ψ 2 (v) and since X(θ u ) has distribution π on [x min , x max ], it follows that E e -α(ηv-θu) = ψ 2 , π /ψ 2 (v) and hence that

E [e -αηv ] = ψ 1 (x new )/ψ 1 (u) • ψ 2 , π /ψ 2 (v).
The ratio in ( 12) can be expressed as

ψ 1 (x new ) ψ 1 (u) • g 1 (u) + ψ 1 (x new ) ψ 1 (u) • ψ 2 , π ψ 2 (v) • g 2 (v) (16)
= E e -αθu g 1 (X(θ u -))I {θu<∞} + e -αηv g 2 (X(η v ))I {ηv<∞} .

Thus the nonlinear optimization problem ( 12) maximizes (2) over the hitting times of the paired process (X, Y ) at thinning and harvesting levels (u, 1) and (v, 2), respectively.

Remark 2. Let d 1 (u) = E e -αθu = ψ 1 (x new )/ψ 1 (u) denote the expected discount factor arising from the rule to thin the forest when (X, Y ) reaches level (u, 1) and let d 2 (v) = E e -α(ηv-θu) = ψ 2 , π /ψ 2 (v) be the expected discount factor associated with the rule to harvest when (X, Y ) hits (v, 2), starting at time θ u in location X(θ u ). The bound (12) then takes the form

d 1 (u)[g 1 (u) + d 2 (v)g 2 (v)].
The first order optimality conditions are therefore

d 1 (u)[g 1 (u) + d 2 (v)g 2 (v)] + d 1 (u)g 1 (u) = 0 d 2 (v)g 2 (v) + d 2 (v)g 2 (v) = 0
which can be rewritten as

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u ∂ ∂u [g 1 (u) + d 2 (v)g 2 (v)] [g 1 (u) + d 2 (v)g 2 (v)] = - ud 1 (u) d 1 (u) vg 2 (v) g 2 (v) = - vd 2 (v) d 2 (v) .
The second equation shows that the optimal harvesting level v * w occurs where the elasticity of the harvest payoff function g 2 equals the negative of the elasticity of the harvesting discount factor d 2 . The optimal thinning level u * w , however, occurs where the partial elasticity of the combined payoffs for thinning g 1 and discounted harvesting d 2 (v)g 2 (v) equals the negative of the elasticity of the thinning discount factor d 1 .

Optimality for the Unrestricted Problem

The results of Section 2.2 determine an optimal pair of thinning and harvesting times (θ * , η * ) ∈ A 1 for the restricted problem. This subsection demonstrates that this pair of decision times is also optimal for the unrestricted problem.

Theorem 4. Assume g 1 and g 2 satisfy Condition 1. Then V w (x new , 1) = V w r (x new , 1) and hence the thinning and harvest times (θ * , η * ) of Theorem 3 is an optimal pair for the unrestricted stochastic forestry problem.

Proof. Choose (θ, η) ∈ A arbitrarily. Select a sequence K n ∞ and define τ Kn = inf{t > 0 :

X(t) = K n }. Since ∞ is a natural boundary point, it follows that τ Kn ∞ as n → ∞. Observe that {(θ ∧ τ Kn , η ∧ τ KN )
} is a sequence of decision rules within the restricted collection A 1 and hence

V w r (x new , 1) ≥ E e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n )<∞} +e -α(η∧τK n ) g 2 (X(η ∧ τ Kn ))I {η∧τK n )<∞} = E e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n )<∞} I {θ<∞} +E e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n )<∞} I {θ=∞} +E e -α(η∧τK n ) g 2 (X(η ∧ τ Kn ))I {η∧τK n <∞} I {η<∞} (17) +E e -α(η∧τK n ) g 2 (X(η ∧ τ Kn ))I {η∧τK n <∞} I {η=∞} .
Consider the first expectation on the right-hand side of ( 17). Observe that on the set where θ is finite, i. e. {θ < ∞}, θ(ω) ∧ τ Kn (ω) = θ(ω) for n sufficiently large. Thus

lim n→∞ e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n <∞} I {θ<∞} = e -αθ g 1 (X(θ))I {θ<∞}
and hence Fatou's lemma implies that

E e -αθ g 1 (X(θ))I {θ<∞} ≤ lim n→∞ E e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n <∞} I {θ<∞} .
Analyzing the second term, notice that on the set {θ = ∞}, θ ∧ τ Kn = τ Kn so

E e -α(θ∧τK n ) g 1 (X(θ ∧ τ Kn ))I {θ∧τK n } I {θ=∞} = E e -ατK n g 1 (X(τ Kn ))I {τK n <∞} I {θ=∞} ≤ g 1 (K n )E e -ατK n I {τK n <∞} = g 1 (K n ) • ψ 1 (x new ) ψ 1 (K n )
and the right-hand side converges to 0 by Condition 1(d). The same analyses applies to the third and fourth expectations in (17) with the result that V w r (x new , 1) ≥ E e -αθ g 1 (X(θ))I {θ<∞} + e -αη g 2 (X(η))I {η<∞} .

Taking the supremum over (θ, η) ∈ A implies V w r (x new , 1) ≥ V w (x new , 1). The reverse inequality follows immediately from the fact that A 1 ⊂ A.

Example

For simplicity, we illustrate the single cycle thinning-and-harvesting problem by looking at the case in which X is a mean reverting process for both dense and thinned forests. The key feature of this model are particular mean levels for dense and thinned forests to which the process is attracted. In our formulation, the mean levels are 1/γ y , with γ y > 0. When the process is smaller than 1/γ y , the drift will be positive, but when the process exceeds this level, the drift will be negative. These levels therefore act as natural targets for the sizes of the dense and thinned forest or for the size of an individual tree in a dense or thinned forest, depending on what the process X models for each type of forest. Specifically, the process X satisfies (1), where for y = 1, 2, μ(x, y) = μ(1γ y x) and σ(x, y) = σ√ x, with μ and σ being fixed positive constants. This model is an example of a process on the state space (0, ∞) for which the left end-point 0 is an entrancenot-exit boundary point. For simplicity, we take the distribution π of the size of the thinned forest stand X(θ) to be a unit point mass at x thin which means X(θ) = x thin following the decision to thin.

The differential operators for the mean-reverting model are

A (y) f (x) = μ(1 -γ y x)f (x) + (σ 2 /2)xf (x), y = 1, 2.
The increasing solutions of the eigenfunction equation ( 5) are given by

ψ y (x) = K M α γ y μ , 2μ σ2 , 2γ y μ σ2 x , ( 18
)
in which K M (a, b, z) denotes the Kummer M -function

K M (a, b, z) = 1 + az b + (a) 2 (b) 2 z 2 2! + • • • + (a) n (b) n z n n! + • • • ,
where for c = a, b, (c) n := c(c + 1)(c + 2) • . . . • (c + n -1), (c) 0 := 1 (see [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] for details). For instance, K M (a, b, z) is a solution of the ordinary differential equation

zf (z) + (b -z)f (z) -af (z) = 0;
an alternative notation for the Kummer M -function is [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF].

K M (a, b, z) = 1 F 1 (a, b; z) in
For illustrative purposes we choose the thinning and harvesting reward functions g 1 and g 2 as follows:

g y (x) = x • δ y • 1+tanh( y (x-zy)) 2 -c y , y= 1, 2, ( 19
)
where for y = 1 and 2, δ y , y , z y and c y are nonnegative constants. Nice interpretations can be given to the parameters: δ y represents the long-term growth rate of timber values (for large values of x) when > 0; the quantity c y may include a common shift in value that applies to all timber sizes (e.g., property tax or transportation costs per tree [START_REF] Davies | Towards more accurate growth simulations and NPV appraisals: Using INFORM to project tree grade and market value increases[END_REF], Table 1); y is a scaling factor and z y is close to the inflection point where profits increase the most. This form for the harvesting payoff function g 2 gives a reasonable family of functions that provide good approximations to the "Tree Value Conversion Standards" (see [START_REF] Mendel | Tree value conversions standards for hardwood sawtimber[END_REF]) which estimate the harvesting payoffs based on the size (diameter at breast height -dbh) and grade of the tree. An example of the fit for sugar maples is provided in Appendix B; appropriate choices of parameters also provide good fits of g 2 for the TVCS of other tree varieties as well. (The tree value conversion standards were revisited in the publication [START_REF] Debald | Tree value conversion standards revisited[END_REF]; we base our illustration on the original paper by [START_REF] Mendel | Tree value conversions standards for hardwood sawtimber[END_REF].) The form ( 19) of payoff function can be used for g 1 to obtain good approximations of the "Fuel Value" (see [START_REF] Morrow | Tree value: A basis for woodland management[END_REF]). Notice that this class of functions includes affine functions when = 0.

The paper [START_REF] Morrow | Tree value: A basis for woodland management[END_REF] lists the TVCS values corresponding to hardwood trees having dbh in the range from 10 inches to 28 inches and comments that trees with dbh in the range of 4 to 10 inches are worth nothing except for fuelwood, but are in the most need of thinning. He further adds that trees with dbh in the range of 10-14 inches have marginal value for timber and the rate of value increase is high, especially for thinned trees. The situation is reversed when the trees have dbh in the range 24 to 28 inches with high values but low growth rates. The function class (19) for g 2 captures this type of value change while appropriate choices of γ y in the mean-reverting stochastic growth model will provide the observed type of growth. Morrow also indicates that trees with dbh of 20 inches "do not just occur; they are the result of thinning young stands, good sites, or a combination of both."

Table 1 illustrates the optimal thinning and harvesting levels u * w and v * w , respectively, for a particular choice of parameters along with the optimal expected discounted payoff V w (x new , 1) for thinning and harvesting. The particular choice of 1 = 0 and z 1 = 0 indicates that a good fit to the fuelwood values is given by an affine function. First observe that the value of g 1 (u * w ) is slightly negative indicating that it is optimal to spend money to thin the forest to receive the benefit of larger and more valuable trees when one harvests. Thus thinning is important in order to develop more valuable trees rather than as a source of income. Notice also the effect of increasing the thinned size of the forest is to thin earlier but this has no impact on the decision to harvest. Intuitively, when x thin is small, the decision to thin would remove larger trees so as to allow smaller, good quality trees to grow to full size and be harvested. When x thin is large, however, the results indicate that the smaller competing trees would be thinned leaving larger trees for later harvest.

These values in Table 1 should be compared with those when thinning is not applied and the process X only evolves according to the mean-reverting regime with parameters μ = 1, σ 2 = 0.03 and 1/γ 1 = 100, and the reward function is given by g 2 . The choice of g 2 for this comparison provides an optimistic value since the trees in a dense forest will not grow to the same size or grade as for a thinned forest, but there will be some trees of sufficient size so that the use of the fuelwood payoff function g 1 would not be appropriate. The 18.8 6 1 .7 5 .526

Table 1: Optimal thinning levels u * w and harvesting levels v * w (in cm) for the Wicksell-Model as well as values of V w (xnew, 1) for various x thin values (in cm) using two mean reverting processes;

μ 1 = μ 2 = 1, σ 2 1 = σ 2 2 = 0.03; 1/γ 1 = 100, 1/γ 2 = 120, α = 0.03, xnew = 0.5, δ 1 = 0.7345, 1 = 0, z 1 = 0, c 1 = 9.1748, δ 2 = 1.8254, 2 = 0.04502, z 2 = 56.6523, c 2 = 4.3862.
optimal harvesting level of the corresponding Wicksell problem equals 58.8 and the optimal value is 4.47. Moreover, the expected harvesting time is ∼ 87 years. For the case with thinning, when x thin = 20 cm or x thin = 22.5 cm in Table 1 (and the associated selection of parameters and payoff functions), for example, the two phases -before thinning and after thinning -have an average length of 22-25 years and 61-64 years respectively. Hence, thinning might slightly increase the average optimal cash flow while the average time up to harvesting is about the same. (The integral formula for the mean hitting time

τ b of a level b > x 0 of a mean-reverting process is E[τ b ] = b x0 v 0 e - R v z 2μ(u,y) σ 2 (u,y) du • 2 σ 2 (z,y
) dz dv, with μ(x, y) and σ(x, y) specified at the beginning of this example (Karlin and Taylor, 1981, pp. 192-197); this formula has been used to determine these mean thinning and harvest times.)

The optimal values u * w , v * w and thinning level x thin serve as recommendations to the forester who would be given the task of determining how to implement thinning and harvesting. Additional comments about the model and implementation are given in Section 4 of concluding remarks.

Faustmann Infinite-Cycle Rotation Problem

In contrast with the Wicksell problem whose goal is to maximize the expected discounted reward for thinning and harvesting a forest one time, the Faustmann problem allows the forest to be replanted after each harvest and therefore rewards are earned over an infinite number of thinning and harvesting cycles.

Reduction to a Wicksell Single-Cycle Problem

This section analyzes the Faustmann problem by using the strong Markov property to relate it to a Wicksell problem for an adjusted payoff function.

We assume the dynamics of the forest growth process X follow (1), that thinning occurs at times {θ k } with the thinned state X(θ k ) having distriubtion π and that harvesting and replanting happens at times η k at which point the process X reinitializes at x new . The Faustmann objective is to maximize (3) over all {(θ k , η k )} ∈ A. Note, we assume θ k ≤ η k ≤ θ k+1 for every k. Recall V f (x new , 1) denotes the optimal value for the Faustmann problem.

To facilitate understanding of the argument for this section, we use a subscript on the expectation operator, e.g. E xnew [•], to indicate the initial position of the process X. This notation is important when using the strong Markov property.

Theorem 5. Assume g 1 and g 2 satisfy Condition 1. Then the optimal value for the Faustmann infinite thinning-and-harvesting cycle problem is characterized by the nonlinear optimization problem

V f (x new , 1) = sup u,v∈F g 1 (u)ψ 2 (v) + ψ 2 , π g 2 (v) ψ 1 (u)ψ 2 (v) -ψ 1 (x new ) ψ 2 , π • ψ 1 (x new ). ( 20
)
Moreover, an optimal pair (u * f , v * f ) exists and the thinning and harvest times are given by the successive hitting times of X to the levels u * f and v * f , namely, setting η * 0 = 0, define, for k = 1, 2, 3, . . . ,

θ * k = inf{t > η * k-1 : X(t-) = u * f } and η * k = inf{t > θ * k : X(t-) = v * f }. ( 21
)
The optimal value is therefore

V f (x new , 1) = g 1 (u * f )ψ 2 (v * f ) + ψ 2 , π g 2 (v * f ) ψ 1 (u * f )ψ 2 (v * f ) -ψ 1 (x new ) ψ 2 , π • ψ 1 (x new ). ( 22
)
Proof. In preparation for the analysis, notice that for k ≥ 2, θ k ≥ η 1 ; define θk-1 = θ kη 1 . Also recall that on the set {η 1 < ∞}, X(η 1 ) = x new and observe that the summands of (3) are 0 for k ≥ 2 on the set {η 1 = ∞}. Then using the strong Markov property in the second last equality below, we have

E xnew ∞ k=1 e -αθk g 1 (X(θ k -))I {θk<∞} = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + E xnew ∞ k=2 e -αθk g 1 (X(θ k -))I {θk<∞} = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + E xnew E xnew e -αη1 I {η1<∞} ∞ k=2 e -α(θk-η1) g 1 (X(η 1 + (θ k -η 1 )-))I {η1+(θk-η1)<∞} F η1 = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + E xnew e -αη1 I {η1<∞} E xnew ∞ k=1 e -α θk g 1 (X(η 1 + θk -))I {η1+ θk<∞} F η1 = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + E xnew e -αη1 I {η1<∞} E X(η1) ∞ k=1 e -α θk g 1 (X( θk -))I { θk<∞} = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + E xnew e -αη1 I {η1<∞} E xnew ∞ k=1
e -α θk g 1 (X( θk -))I { θk<∞} .

Define ηk-1 = η kη 1 , for k ≥ 2. The same analysis can be applied to the rewards obtained from harvesting and thus

E xnew ∞ k=1 e -αθk g 1 (X(θ k -))I {θk<∞} + e -αηk g 2 (X(η k -))I {ηk<∞} = E xnew ⎡ ⎣ e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + e -αη1 g 2 (X(η 1 -))I {η1<∞} (23) + e -αη1 I {η1<∞} E xnew ∞ k=1 e -α θk g 1 (X( θk -))I { θk<∞} + e -αηk g 2 (X(η k -))I {ηk<∞} .
Denote the infinite sequences by ( θ, η) = {( θk , ηk )} and notice ( θ, η) ∈ A. Observe

V f (x new , 1) = sup ( θ,η)∈A E xnew ∞ k=1
e -α θk g 1 (X( θk -))I { θk<∞} + e -αηk g 2 (X(η k -))I {ηk<∞} .

Taking the supremum of the right-hand side of ( 23) over ( θ, η) ∈ A implies

E xnew ∞ k=1 e -αθk g 1 (X(θ k -))I {θk<∞} + e -αηk g 2 (X(η k -))I {ηk<∞} ≤ E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + e -αη1 g 2 (X(η 1 -))I {η1<∞} + e -αη1 I {η1<∞} V f (x new , 1) = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + e -αη1 [g 2 (X(η 1 -)) + V f (x new , 1)]I {η1<∞} = E xnew e -αθ1 g 1 (X(θ 1 -))I {θ1<∞} + e -αη1 g2 (X(η 1 -))I {η1<∞}
where g2 (x) = g 2 (x) + V f (x new , 1). Thus the Faustmann infinite-cycle rotation problem is bounded above by a Wicksell single-cycle problem using the thinning payoff function g 1 and a shifted harvesting payoff function g2 . In a similar manner, begin by taking the supremum of the left-hand side of ( 23) over (θ, η) ∈ A, then the supremum of the right-hand side over ( θ, η) ∈ A and finally the supremum of the right-hand side over (θ 1 , η 1 ) in the Wicksell admissible set to obtain the opposite inequality and hence the value V f (x new , 1) equals the value Ṽ w (x new , 1) of this Wicksell problem. Theorem 3 establishes existence of optimizers u * f and v * f for which the value of this Wicksell problem (and hence the Faustmann problem) is 24) and solving for V f (x new , 1) yields

V f (x new , 1) = Ṽ w (x new , 1) = g 1 (u * f )ψ 2 (v * f ) + ψ 2 , π g2 (v * f ) ψ 1 (u * f )ψ 2 (v * f ) • ψ 1 (x new ). ( 24
) Substituting g2 (v * f ) = g 2 (v * f ) + V f (x new , 1) into (
V f (x new , 1) = ψ 1 (x new )[g 1 (u * f )ψ 2 (v * f ) + ψ 2 (x thin )g 2 (v * f )] ψ 1 (u * f )ψ 2 (v * f ) • 1 - ψ 1 (x new ) ψ 2 , π ψ 1 (u * f )ψ 2 (v * f ) -1 = g 1 (u * f )ψ 2 (v * f ) + ψ 2 (x thin )g 2 (v * f ) ψ 1 (u * f )ψ 2 (v * f ) -ψ 1 (x new ) ψ 2 , π • ψ 1 (x new ). (25)
Now define (θ, η) ∈ A to be the successive hitting times of u * f and v * f defined in (21). Then

X(θ * k -) = u * f and X(η * k -) = v *
f and the evaluation of (3) then involves determining the value of the infinite series comprised of the Laplace transform of the successive hitting times of the levels u * f or v * f . The evaluation of these series is given in Appendix C with the result that the expected discounted reward obtained using this policy achieves the bound (25).

The final point to be addressed is the demonstration that (u * f , v * f ) is an optimizing pair for the nonlinear optimization problem (20). To see this, the equality of V f (x new , 1) and Ṽ w (x new , 1) in (24) implies that

V f (x new , 1) = sup u,v∈F g 1 (u)ψ 2 (v) + ψ 2 , π g2 (v) ψ 1 (u)ψ 2 (v) • ψ 1 (x new )
and hence for each u, v ∈ F ,

V f (x new , 1) ≥ g 1 (u)ψ 2 (v) + ψ 2 , π g2 (v) ψ 1 (u)ψ 2 (v) • ψ 1 (x new ) = g 1 (u)ψ 2 (v) + ψ 2 , π g 2 (v) ψ 1 (u)ψ 2 (v) • ψ 1 (x new ) + ψ 1 (x new ) ψ 2 , π ψ 1 (u)ψ 2 (v) • V f (x new , 1). Solving for V f (x new , 1) yields for each u, v ∈ F , V f (x new , 1) ≥ g 1 (u)ψ 2 (v) + ψ 2 , π g 2 (v) ψ 1 (u ψ 2 (v) -ψ 1 (x new ) ψ 2 , π • ψ 1 (x new )
with equality for (u * f , v * f ), establishing (20).

Remark 3. A similar remark to that of Remark 2 holds for the Faustmann problem. Specifically, let

d 1 (u, v) = ψ 1 (x new )ψ 2 (v) ψ 1 (u)ψ 2 (v) -ψ 1 (x new ) ψ 2 , π and d 2 (u, v) = ψ 1 (x new ) ψ 2 , π ψ 1 (u)ψ 2 (v) -ψ 1 (x new ) ψ 2 , π
denote the sums of the expected thinning discount factors and the expected harvesting discount factors, respectively, when thinning occurs at level u and harvesting occurs at level v. Observe the nonlinear function (20) to be maximized takes the form 20) can be written as d 2 (u, v)g (u, v). The first order optimality conditions for (u

d 1 (u, v)g 1 (u) + d 2 (u, v)g 2 (v). Also note that d 1 (u, v) = ψ2(v) ψ2,π • d 2 (u, v). Letting g(u, v) = ψ2(v) ψ2,π • g 1 (u) + g 2 (v), (
* f , v * f ) yield ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u * f ∂g ∂u (u * f , v * f ) g(u * f , v * f ) = - u * f ∂d2 ∂u (u * f , v * f ) d 2 (u * f , v * f ) v * f ∂g ∂v (u * f , v * f ) g(u * f , v * f ) = - v * f ∂d2 ∂v (u * f , v * f ) d 2 (u * f , v * f ) and since ∂d1 ∂u (u, v) d 1 (u, v) = ∂d2 ∂u (u, v) d 2 (u, v)
it follows that at a pair (u * f , v * f ) of optimizers, the partial elasticities of the payoff function g equal the negative of the partial elasticities of the discount factors:

EL u [g](u * f , v * f ) = -EL u [d 1 ](u * f , v * f ) EL v [g](u * f , v * f ) = -EL v [d 2 ](u * f , v * f )
Notice that the payoff function g is valued in currency at the time of harvesting, not its present-value.

Example

The illustration of the Faustmann solution uses the same model formulation as for the Wicksell problem. Recall the growth process X satisfies (1) with μ(x, y) = μ(1γ y x) and σ(x, y) = σ√ x, where μ, σ and γ y are fixed positive constants, while each thinning decision produces a value of X(θ k ) = x thin and harvesting restarts the forest value at X(η k ) = x new . Also recall that the differential operators for the mean-reverting model are A (y) f (x) = μ(1γ y x)f (x) + (σ 2 /2)xf (x), y = 1, 2, the increasing solutions of the eigenfunction equation ( 5) are given by ( 18) and the payoff functions are as in (19).

Table 2 displays the optimal thinning and harvesting levels u * f and v * f along with the optimal payoff for the Faustmann problem. Comparing these values with Table 1, one observes the effect of optimizing over infinitely many cycles reduces both the optimal thinning and harvesting levels while at the same time increases the payoff received. Moreover, these optimal thinning and harvesting levels are in agreement with the comments by [START_REF] Morrow | Tree value: A basis for woodland management[END_REF] on the need to thin a stand of hardwoods when dbh is in the 4 to 10 inch range and the valuation of the trees is high when dbh is 24 to 28 inches (∼ 60 cm to ∼ 70 cm). 16.1 5 9 .9 6 .249

xthin u * f v * f V f (xnew,
Table 2: Optimal thinning levels u * f and harvesting levels v * f (in cm) for the Faustmann-Model as well as values of V f (xnew, 1) for various x thin values (in cm) using two mean reverting processes; μ 1 = μ 2 = 1, σ 2 1 = σ 2 2 = 0.03; 1/γ 1 = 100, 1/γ 2 = 120, α = 0.03, xnew = 0.5, δ 1 = 0.7345, 1 = 0, z 1 = 0, c 1 = 9.1748, δ 2 = 1.8254, 2 = 0.04502, z 2 = 56.6523, c 2 = 4.3862.

Concluding Remarks

This paper has examined a variation of the Wicksell single cycle and Faustmann on-going harvest rotation problems that includes thinning of the forest to promote better growth dynamics. The model adopts a deterministic pricing function. This choice can be understood to be derived from aggregate data over a long-term horizon and thus the problem under consideration treats rotation management as a long-term investment. One might use this problem to determine an approximate age or size of tree at which harvesting would occur. Then when one approaches the harvesting and marketing of the forest, one would take a short-term view and use a stochastic pricing model to decide on the precise harvesting time.

The analysis in this paper replaces the stochastic model by an infinite-dimensional linear program over a space of (deterministic) measures. This approach proved to be quite tractable for the problems under study in this paper. An interesting feature of this analysis is the reduction of the Faustmann infinite-cycle problem to a Wicksell single-cycle problem through the use of the strong Markov property for the growth process.

The results for the thinning-and-harvest problems immediately reduce to the known results for the harvest only problems. When there is no change in dynamics allowed, there is only one strictly increasing solution to the eigenvalue problem Af = αf and there would be no dependence on a Y process. For the Wicksell problem, one eliminates the g 2 term from the optimization problem (12) since this would provide a second harvest opportunity. Theorem 3 then reduces to the known solution (see, e.g., [START_REF] Sødal | The stochastic rotation problem: A comment[END_REF]). The reduction for the Faustmann problem is even more immediate. When harvesting is the only decision to make, there would be a single payoff function g, only one reinitializing point x new , one decision level u * and as above only one increasing function ψ. The expression (22) in Theorem 5 then simplifies to the known result.

This methodology can be easily adapted to include additional features to the model. For instance, the owner of the forest stand may receive a running payment stream that depends on the size of the forest; such payments might represent the amenity value of the forest or carbon credit payments that are received as long as the forest is allowed to grow (see [START_REF] Helmes | Construction of the value function and stopping rules in optimal stopping of onedimensional diffusions[END_REF] for a single-cycle example using this methodology). The dynamics may also include sudden destruction due to fire or pests. Assuming the occurrences of such destruction are modelled by a Poisson process with some probability distribution on the size of the forest following the occurrence, the methods of this paper apply with only a minor modification to the generator A and an adjustment to the discount factor α.

Proposition 1. Define the functions H(x, y) = F 1 (x)G 2 (y) + G 1 (x)F 2 (y) and h(x, y) = g 1 (x)G 2 (y) + G 1 (x)g 2 (y) and assume H is strictly positive. Then an upper bound on the optimal value of (A.1) is given by g 1 (x) ν 1 (dx) + g 2 (y) ν 2 (dy) ≤ sup (x,y) h(x, y) H(x, y) and if the function h/H has a global maximum, then this bound is achieved.

Proof. Let (ν 1 , ν 2 ) be a feasible pair of measures and note that at least one (hence both) measures have positive mass. Let ν = ν 1 × ν 2 be the product measure on (E 1 × E 2 , F 1 × F 2 ). Observe that

H dν = F 1 dν 1 G 2 dν 2 + G 1 dν 1 F 2 dν 2 = F 1 dν 1 + F 2 dν 2 G 2 dν 2
where the second constraint has been used and hence

H G 2 dν 2 dν = 1.
Thus a probability measure ν can be defined by taking H/( G 2 dν 2 ) to be the density with respect to ν. Now observe that

h dν = g 1 dν 1 G 2 dν 2 + G 1 dν 1 g 2 dν 2 = g 1 dν 1 + g 2 dν 2 G 2 dν 2 .
So as a result,

g 1 dν 1 + g 2 dν 2 = h G 2 dν 2 dν = h H • H G 2 dν 2 dν = h H dν.
For any probability measure ν, the value of the objective is bounded above by sup x,y h(x,y) H(x,y) . Moreover, when the ratio h/H achieves its maximum, say at (x * , y * ), an optimal pair of measures (ν * 1 , ν * 2 ) can be determined by taking ν = δ {(x * ,y * )} . This then implies that the corresponding product measure ν * also places a point mass on {(x * , y * )} having mass R G2 dν2

H(x * ,y * ) and moreover that the measure ν * 1 is a point mass on {x * } and similarly ν * 2 is a point mass on {y * }. Now utilizing the two constraints of the linear program, we are able to determine the masses of ν * 1 and ν * 2 from the system of equations Similarly, let θk = θ kη k-1 , note θk is the first hitting time of X (with Y (t) = 1) started at time η k-1 in location x new and E[e -α θk ] = ψ 1 (x new )/ψ 1 (u). Applying the strong Markov property, we have α[ηk-1+(θk-ηk-1)] = E xnew e -αηk-1 E xnew e -α(θk-ηk-1) F ηk-1

F 1 (x * ) ν * 1 {x * } + F 2 (y * ) ν * 2 {y * } = 1 G 1 (x * ) ν * 1 {x * } -G 2 (y * ) ν * 2 {y * } = 0 yielding ν * 1 {x * } = G 2 (y * ) F 1 (x * )G 2 (y * ) + G 1 (x * )F 2 (y * ) , ν * 2 {y * } = G 1 (x * ) F 1 (x * )G 2 (y * ) + G 1 (x
E xnew e -αθk = E xnew e -
= E xnew e -αηk-1 E X(ηk-1) e -α θk (C.2)

= E xnew e -αηk-1 • ψ 1 (x new ) ψ 1 (u) .

Iterating (C.1) and (C.2) determines the summands of each series and hence yields

E ∞ k=1 e -αηk = ∞ k=1 ψ 1 (x new ) ψ 1 (u) • ψ 2 , π ψ 2 (v) k = 1 1 -ψ1(xnew) ψ2,π ψ1(u)ψ2(v) • ψ 1 (x new ) ψ 2 , π ψ 1 (u)ψ 2 (v) = ψ 1 (x new ) ψ 2 , π ψ 1 (u)ψ 2 (v) -ψ 1 (x new ) ψ 2 , π and E ∞ k=1 e -αθk = ∞ k=1 ψ 1 (x new ) ψ 1 (u) k • ψ 2 , π ψ 2 (v) k-1 = ψ 2 (v) ψ 2 , π ∞ k=1 ψ 1 (x new ) ψ 1 (u) • ψ 2 , π ψ 2 (v) k = ψ 1 (x new )ψ 2 (v) ψ 1 (u)ψ 2 (v) -ψ 1 (x new ) ψ 2 , π .

X

  (t-) = u} and η k = inf{t ≥ θ k : X(t-) = v}. We seek to determine the expressions for the expected discount factorsTo determine E [e -αηk ] define ηk = η kθ k and observe that ηk gives the first hitting time of level v of the process X under the dynamics with Y (t) = 1 starting at time θ k in location X(θ k ) having distribution π on [x min , x max ] independent of θ k . Recall from Remark 1, the Laplace transform of ηk is given by ψ 2 , π /ψ 2 (v). Using the strong Markov property and subscripts on the expectation operator to indicate the initial value of X, we haveE xnew e -αηk = E xnew e -α[θk+(ηk-θk)]= E xnew e -αθk E xnew e -α(ηk-θk) F θk = E xnew e -αθk E X(θk) e -αηk (C.1) = E xnew e -αθk E E X(θk) e -αηk = E xnew e -αθk • ψ 2 , π ψ 2 (v) .

  )F 2 (y * ) . Table B.3: Typical Change in Size and Value with Growth of a Sugar Maple

	dbh	Tree Value Conversion Standards
	(in cm)	(TVCS)
	25	0
	30	1
	35	3
	40	8
	45	17
	50	29
	60	58
	70	94

*

Acknowledgement

The authors would like to thank Hans Volkmer for the simple proof of the optimization result in Proposition 1.

1 This research has been supported in part by the U.S. National Security Agency under Grant Agreement Number H98230-09-1-0002. The United States Government is authorized to reproduce and distribute

Appendix A. Optimization of Linear Program (11)

In this appendix, we prove that an optimal choice of measures for (11) will place point masses on locations which maximize (12). To do so, however, we phrase the problem more generally as one of finding finite measures ν 1 and ν 2 so as to solve the linear programming problem

in which F i are non-negative, measurable, G i are positive, measurable and g i are measurable functions on measurable spaces (E i , F i ), i = 1, 2. To be feasible, the measures ν 1 and ν 2 must be such that for i = 1, 2, F i and G i are integrable with respect to ν i .

The form of the harvesting payoff function

2 -c 2 is chosen since the parameters δ 2 , 2 , z 2 and c 2 provide enough flexibility to accurately capture pricing data. For example, [START_REF] Morrow | Tree value: A basis for woodland management[END_REF] reports pricing data (based on 1972 prices) over their growth cycle for a number of different varieties of trees. Table B.3 displays the data for the growth of a sugar maple; dbh stands for the diameter at breast height. Tree Value Conversion Standards (TVCS) provide a measure of a tree's worth, based on the comparative value of the quantity and quality of expected yield of one-inch lumber, taking into account conversion costs such as harvesting, transporting, and milling. It is a standard by which trees of different sizes can be compared, but it excludes most price effects of inflation and the marketplace. A fit of this pricing data using the parameters δ 2 = 1.8254, 2 = 0.04502, z 2 = 56.6523 and c 2 = 4.3862 is displayed in Figure B.1. We can see this family of pricing functions provides an excellent fit to the data; very good fits are also provided with an appropriate change in the parameters for the prices corresponding to other varieties of trees.

Appendix C. Expected Discount Factors

Consider the thinning-and-harvesting rule whereby a dense forest is thinned whenever it reaches level u and is harvested whenever a thinned forest achieves size v. Recall, the initial state of the forest is (X(0), Y (0)) = (x new , 1) indicating that the stand is dense and new. Define the "zeroeth" harvesting time η 0 = 0. Now define the successive thinning and harvest times (for k = 1, 2, 3, . . .) by θ k = inf{t ≥ η k-1 :


Reinhard Stößer

Michael Feist
email: feistm@chemie.hu-berlin.de

Kerstin Patzwaldt

Michael Menzel

Franziska Emmerling
email: franziska.emmerling@bam.de

Redox reactions of K 3 [Fe(CN) 6 ] during mechanochemically stimulated phase transitions of AlOOH

Keywords:

Thermally induced redox reactions of K 3 [Fe(CN) 6 ] (1) were investigated for a broad temperature range by thermal methods and structure analytical methods (ESR and Mößbauer spectroscopy, Xray Powder diffraction and XANES). Based on the influence of the mechanically activated and transforming matrices 2 and 3, redox processes can be tuned to form doped Al 2 O 3 systems which contain either isolated Fe 3+ centres or redox active phases and precursors like (Al 1-x Fe x ) 2 O 3 (4), (Al 3-x Fe x )O 4 (5), Fe 3 O 4 , Fe 2 O 3 and Fe 0 . The phase Fe 3 C and the chemically reactive C-species were detected during the reaction of 1. The final composition of the doped products of D-Al 2 O 3 is mainly influenced by the chemical nature of the Fe doping component, the applied temperature and time regime, and the composition of the gas phase (N 2 , N 2 /O 2 or N 2 /H 2 ). From the solid state chemistry point of view it is interesting that the transforming matrix (2, 3) possesses both oxidative and protective properties and that the incorporation of the Fe species can be performed systematically.

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Introduction

Potassium hexacyanoferrate(III), K 3 [Fe(CN) 6 ] (1), continues to be a chemical "evergreen" which is also true for Berlin Blue. This could be deduced, last but not least, from the observation of a never ending stream of reports on novel chemical and application aspects found for 1. The present paper contributes to this by reporting some unexpected spectroscopic and chemical findings which were obtained by a combination of thermal and spectroscopic investigations of 1 in the form of the pure substance as well as after mechanochemically performed doping of AlOOH xerogels (2) with 1 and other Fe species, followed by a thermal treatment.

The aim of this study is deduced from the question whether a targeted Fe doping of Al 2 O 3 samples could be realized by combining the competing mechanochemically and thermally stimulated redox processes of iron with the transformation processes occurring within an AlOOH/Al 2 O 3 matrix. This question is closely related to the preparation of supported aluminium oxide catalysts with a defined Fe II or Fe III content, as well as to the preparation of magnetic materials [1][2][3] by primarily employing the chemical properties of the Al 2 O 3 precursors and the special properties of their AlO x matrices.

A first and important step concerning the investigation of the thermal decomposition of 1 was given by Wolski et al. [4][5]. Based on careful chemical analysis and long term thermal analysis, the authors determined the structures of several intermediates. The investigations described here support the results of the authors concerning pure 1, but include additional findings. The synthesis and structural characterisation of the pure crystalline phases of ternary and quaternary metal nitrides [6], carbides [7] and carbometallates [8] and their thermochemistry [9] is well documented in the literature and contain numerous novel and unique results. The approach is comparable to that followed in the current publication. Here, however, we focused on a more phenomenological description with regard to the redox processes of 1 competing with the chemistry of the AlO(OH) matrix transformation.

Results and Discussion

The investigation of the thermal behaviour of pure 1 by employing a thermal analyzer simultaneously coupled to a mass spectrometer (TA-MS) is depicted in Figure 1a together with the results of the characterization of separately tempered (0.5 h/N 2 ) samples corresponding to the essential stages of the thermonanalytical (TA) curves obtained during the linear heating run. The thermal activation causes an internal redox process in 1 that results in the liberation of (CN) 2 and to the formation of lower-valent Fe species. In this concern, the good agreement of the TA-MS findings (Figure 1a) to the Mößbauer data is obvious. They were confirmed and completed by the results of the X-ray powder diffraction (XRD), the X-ray absorption near edge spectroscopy (XANES), and the electron spin resonance (ESR) spectroscopy (Supporting information in S1 -S5). Stable intermediates being formed in defined temperature-time ranges could be identified (Figures 1b,1c) and allow description of the evolution of the Fe 3 C and Fe 0 product phases. The thermal process is accompanied by colour changes starting with yellow, followed by greenishblue to black. Furthermore, one can deduce from Figure 1 that the thermally initiated reactions start above 250 °C and that the well separated mass loss around 300 °C is caused by the exothermal (peak temperature T P 338 °C) release of (CN) 2 and, to a minor extent, of HCN. This is accompanied by the formation of Fe II species in the solid residue, which is clearly indicated by the results of the Mößbauer investigations together with XRD and XANES (cf. S2 -S4). The XANES measurements were especially helpful in the case of sample 1 tempered at 300 °C. Here, the formation of Fe II was indicated by the Mößbauer data, but could not be confirmed by XRD due to a low crystallinity of the solid phase formed. It could be shown by XANES, however, that 21 % Fe II was contained in the sample with, expectedly, a major part of Fe III (cf. S4). It is obvious that structurally distorted precursors of Berlin Blue were formed [10][START_REF] Verdaguev | Magnetism: Molecules and Materials[END_REF]. In the following temperature range between 400 and 650 °C, however, significant changes could not be established neither by thermal nor by spectroscopic methods. In the solid residues, K 4 [Fe(CN) 6 ] is contained, as could be evidenced by XRD. Above 700 °C, on the other hand, Fe 3 C is unambiguously detected both by Mößbauer spectroscopy and XRD, even if K 4 [Fe(CN) 6 ] remains to be the major component of the product phase.

Above 900 °C, metallic Fe 0 can be detected unambiguously in the solid residue (Figure 1b,c); it is coexisting with KHCO 3 (cf. S3a), which is unusual, but in line with the mass spectrometric detection of CO 2 . This formation of carbonate or CO 2 can be explained by highly reactive carbon formed within the decomposing matrix of 1 and is obviously able to react with oxygen traces in the carrier gas. 1After the final tempering of 1 under N 2 at 1100 °C, the phases D-Fe und Fe 3 C (Cohenite) could be detected by XRD which directly supports the Mößbauer findings (Figure 1, cf. also S2c) and is in line with the ESR data, i.e. the ferromagnetic resonances.

As a mechanism for this kind of "self-reduction" of 1, a reaction sequence can be proposed that begins with the release of (CN) 2 , is then followed by transformations yielding Fe II 3 (CN) 6 , and ends up with the formation of D-Fe, Fe 3 C, and C (Figure 2). This would be both a support and a substantial extension of the decomposition mechanism proposed by Raj and Danor [START_REF] Raj | [END_REF].

A completely different decomposition proceeds if untreated 1 is mixed with a mechanically activated (50 h high-energy ball milling), originally crystalline AlOOH (3), then co-milled for further 8 h, and subsequently subjected to a TA heating run (Figure 3a). The TA-MS curves of the mixture ^30% 1 + 70% 3` and the Mößbauer spectrum of the residue after 1200 °C under N 2 in Figure 3 clearly show that the intermediates formed from 1 react with the transforming matrix 3. A certain similarity with the TA-MS curves in Figure 1 might be deduced, but undoubtedly no (CN) 2 is formed here and, as before, a low quantity of HCN (200-500 °C) is released. Furthermore, the water release from 3 starts earlier (ca. 160 °C) than in the case of matrix 2, but is intense and spread over a large temperature range up to 500 °C as does HCN.

The Mößbauer spectrum of the residue of the TA run comprises two sextets caused by 62.1 % Fe 0 and 15.4 % Fe 3 C, respectively, and one singlet (22.5 %) showing a comparably small isomer shift (IS) that has to be related to Fe III (cf. S2). This Fe III species was formed by an oxidative reaction of 1 with the matrix 3, even if internal reduction processes of 1 are still predominating in this concentration range (30% 1).

If the concentration of 1 in the mixture with matrix 3 is lowered, e.g. down to 1 wt-% such as when applied for the study on the transformation of AlOOH xerogels yielding D- [13], the reactive interaction of the intermediates with the matrix is more strongly expressed. The formation of Fe 3 C or Fe 0 under N 2 , for instance, is not detectable any more.

Al 2 O 3
Figure 4 depicts schematically the experimental approach to yield the mentioned results.

From a mechanochemical point of view it is of particular importance to know that the influence of doping with 1 is weak, even negative concerning the down-shift of the crystallization temperature T P (formation of D-Al 2 O 3 ). Compared with the average value of the T P decrease of 10.9 % (see Table 1), one reaches here only ~20 % of the activation effect being related to the undoped matrix. This means that the crystallization starting from the surface of the activated xerogels is blocked by 1 and its decomposition products. This effect, which might be regarded as a mechanochemical deactivation, could be achieved with other dopants as well (cf. S5) which were unable to form appropriate crystallization seeds such as, for example, Fe 2 O 3 . Up to now, this kind of a netto activation effect is expressed in the strongest manner in the case of 1 and has to be attributed to its specific thermochemical behaviour, including the reaction with the matrix.

The investigation of several reference systems (Figure 5) allowed it to be demonstrated that (i) the results presented here can be generalized to some extent, (ii) which kind of redox processes occur in the systems, and (iii) how these redox reactions can be influenced by the chemical nature of the Fe 3+ dopant. 4) and (Al 3-x Fe x )O 4 (5) [14], symbolized in Figure 5 by Fe 3+ or Fe 2+ respectively, is possible due to competing processes initiated by the type of Fe ligand, the oxidative power of the transforming AlOOH matrix, as well as the chosen gas atmosphere in the TA runs (cf. especially S1b, integrals in the range of 330 mT).

As emphasized by

Interesting results were obtained after the doping of 2 with Fe 3 O 4 , Fe 2 O 3 , or Fe 0 . In this concern, the comparably large variation of the Fe 0 fraction of the total Fe content in the product D-Al 2 O 3 is noteworthy. This fraction decreases down to ~50 % in the case of reactive interaction with the matrix, whereas one obtains 100 % Fe 0 for pure 1 under N 2 /10%H 2 . If the internal oxidative processes were reinforced by employing Fe(NO 3 )9H 2 O, despite the forming gas atmosphere, N 2 /10%H 2 , the Fe 0 content can be lowered down to ~40 %.

Finally the addition of Fe 0 powder to 2, proves that, indeed, the oxidation during interaction with the transforming matrix 2 proceeds stepwise. This is a further support of the proposed interpretation of the findings obtained for the systems ^1/Matrix`. In fact, the formation of 59 % Fe II and even 21 % Fe III due to oxidation by the matrix (i.e. by in situ liberated water) could be observed after co-milling of 2 with 1 wt-% Fe 0 and subsequent TA heating run under N 2 . Besides these Fe II or Fe III species, protected inside the matrix in the form of the phase precursors 4 and 5, the coexisting phase Fe 3 O 4 being detected exclusively under these conditions was formed as an alternative to the incorporation in the Al-rich phases.

The mechanochemical reactions of the matrix are not restricted to thin layers of the surface region only. XRD studies on mixtures of pre-heated samples of 1 (30 min at 200, 400, 700, 900 °C under N 2 , respectively) and 3 demonstrated that the formation of structurally distorted corundum can be evidenced already before tempering (cf. ESR results in S1) which means that parts of the bulk matrix were transformed mechanochemically. Corundum particles formed this way are capable to act as seeds for the crystallization of the entire sample and hereby influences the peak temperature T P of the DTA effect as well (see Table 1). This is in line with the observed ESR signals at g'~4.3 [15] which, during the subsequent tempering step, change into the typical Fe III -ESR fine structure for D-Al 2 O 3 :Fe 3+ with g'~13 and 5.2 [13,[16][17] thus indicating a more pronounced ordering of the structure. Furthermore, the protective (i.e. trapping) function of the matrix for the incorporated Fe II and Fe III species could be evidenced by the ESR data (cf. S1).

Conclusions

Experimental

All chemicals were of analytical purity (p.A.) and used as commercially available: K 3 [Fe(CN) 6 ] (Laborchemie Apolda, Germany), AlOOH (Nabaltec, USA).

The thermal behaviour was investigated by employing a simultaneously coupled TA-MS device (Netzsch STA 409 C Skimmer with a mass spectrometer Balzers QMG 421) in various gas atmospheres (Details in S5). The mechanical activation was performed using a planetary ball mill Pulverisette (Fritsch, Germany; details in S7). The ESR spectra were recorded on an X-Band spectrometer (ZWG-Magnettech GmbH, Berlin-Adlershof, Germany) with MgO:Cr 3+ as a standard both for intensity and g' values (g'=1,9796). The Mößbauer spectra were obtained with a Wissel Electronics spectrometer (Starnberg, Germany) at 295 and 77 K, respectively. All spectra were normalized to the standard D-Fe foil (GoodFellow, USA, details in S2). Details about the XRD and XANES measurements can be taken from S3 and S4.

Received:

Published online:

Figure 2

Simplified reaction scheme of the redox processes occurring in in the systems K 3 [Fe(CN) 6 ], K 3 [Fe(CN) 6 ] / AlOOH und Fe 0 / AlOOH. The reaction sequence supports and extends the mechanism proposed in [START_REF] Raj | [END_REF].

Figure 4

Mechanical activation and thermal treatment of the samples of K 3 [Fe(CN)] 6 (1) and AlOOH (2) in different gas atmospheres

Figure 5

Reference systems: Co-milling of AlOOH xerogels (2) with 1 wt-% doping by different iron compounds as pretreatment for the TA heating runs up to 1200 °C in various gas atmospheres. The phase compositions of the samples after heating comprise compounds of Fe 0 , Fe 2+ , and Fe 3+ in different structural environments, among them magnetite (M) and hematite in two forms (H a , H b ). The Fe content is governed both by gas atmosphere and chemical nature of the dopant.

The formed AlO x matrix exhibits both oxidative and protective properties.

Figure 5 ,

 5 promising possibilities exist for the reactive doping of Al 2 O 3 matrices. The surface enrichment of the D-Al 2 O 3 crystallites with the phase precursors (Al 1-x Fe x ) 2 O 3 (

  Summarizing the presented findings, it should be emphasized that they represent (a) the evidence of intermediates of the thermal transformations of 1, (b) the demonstration of both the oxidative and protective function of the thermally transforming matrix 2, together with (c) the oxidative and disperse incorporation of Fe 3+ ions into the forming D-Al 2 O 3 matrix, (d) the proof of the coexistence of phases with various desired oxidation states of iron, and finally, (e) the demonstration of netto effects causing the decrease of the crystallization temperatures of D-Al 2 O 3 after mechanical activation of mixtures of 2 with Fe III or Fe 0 species. The choice of the doping component together with an appropriate temperature-time regime, as well as the required composition of the gas phase during the TA heating runs allows either distinct product phases originating from the doping component with particular local magnetical properties (i.e. Fe 0 , Fe 2 O 3 , Fe 3 O 4 , etc.) to be obtained or to force the incorporation of Fe x+ into the phase precursors such as 4 or 5 up to the x-Al 2 O 3 phases (x=D,J) which would be a result of the surface-constituting influence of the transforming AlO x matrix.

Figure captions Figure 1 ( 2 +

 12 Figure captions

Figure 3 ( 2 +)

 32 Figure 3

Figure

  Figure(s)

Table 1

 1 Peak temperatures T P (in °C) of the crystallization peaks of D-Al 2 O 3 in the DTA heating curves in different gas atmospheres together with the percentage of decrease 'T P as a result of the milling or co-milling of the AlOOH xerogel with different dopants (1 wt-% each) Mechanically activated K 3 [Fe(CN) 6 ] /AlOOH forms Fe-doped AlO x phases after heating.

	T P / ' T P

Measurements with an oxygen sensor (Dr. J. Hanss, University of Augsburg, Germany) yielded an oxygen content in the inert gas (Ar, N

) of ca. 80 ppm. A comparing TA-MS measurement of non-milled Fe powder under N 2 and the same conditions as applied for 1 exhibited no mass gain at all, i.e. oxidation does not occur under these conditions.


Julien Gony

Matthieu Cord

Sylvie Philipp-Foliguet

Philippe H Gosselin

Frederic Precioso

Michel Jordan

RETIN: a Smart Interactive Digital Media Retrieval System

Keywords: H.3.1 [Information Storage And Retrieval]: Content Analysis and Indexing-Indexing methods; H.3.3 [Information Storage And Retrieval]: Information Search and Retrieval-Retrieval models, Relevance feedback Machine Learning, Content-Based Retrieval, Multimedia

This demonstration presents a digital media retrieval system for searching large categories in different media databases. The core of our system is an interactive online classification based on user labeling. The classification is obtained with a statistical learning method: kernels for similarity representation and SVM (Support Vector Machine) using binary user annotations. RETIN applies also an active learning strategy for proposing documents to the user for labeling. The system can deal with images, 3D objects and videos and other media can be added to. A graphical user interface allows easy browsing of different media, simple and user-friendly interaction and fast retrieval.

INTRODUCTION

Human interactive systems have attracted a lot of research interest in recent years, especially for content-based retrieval systems. Traditional techniques in CBR are limited by the semantic gap, which separates the low-level information extracted from the document and the semantic user request.

To fulfil this semantic gap, interactive systems ask the user to conduct search within the database. Starting with a coarse query, the interactive process allows the user to refine the query as much as necessary. The user interaction consists in binary labels indicating whether or not the document belongs to the desired category (relevance feedback). The main idea of relevance feedback is to use information provided by the user to improve system effectiveness.

In category search, each document has to be classified as belonging or not to the category. The objective of the statistical methods is to update a relevance function or a binary classification of document using the user labels. Discrimination methods (from statistical learning) may significantly improve the effectiveness of visual information retrieval tasks. This approach treats the relevance feedback problem as a supervised learning problem. A binary classifier is learnt by using all relevant and irrelevant labeled images as input training data.

The RETIN search engine, developed in ETIS lab, addresses the problem of category search. Our strategy focus on the online retrieval step to reduce the semantic gap. This is achieved with statistical learning techniques applied on media document category retrieval. RETIN is based on kernels functions to represent similarity, Support Vector Machines for classification and on an active learning strategy to boost the training step. See [START_REF] Cord | Stochastic exploration and active learning for image retrieval[END_REF], [START_REF] Gosselin | A comparison of active classification methods for content-based image retrieval[END_REF], [START_REF] Gosselin | Feature based approach to semi-supervised similarity learning[END_REF] for more informations.

RETIN SYSTEM OVERVIEW

RETIN is composed of two parts. The first one is the offline indexing of the database, where we compute different descriptors for each media, and store the results in an index database. The second one is the online search which is composed of a graphical user interface for query, results display and user labeling, and a search engine using an active learning strategy method. RETIN scheme is shown on figure 1.

RETIN Interactive retrieval

The system uses an active learning scheme based on binary classification in order to interact with a user looking for documents concepts in databases. The scheme is summarized in figure 2. More details can be found in [START_REF] Cord | Stochastic exploration and active learning for image retrieval[END_REF].

Initialization

A retrieval session is initialized from one document brought by the user. The features are computed on that new document and added to the database. This document is then labeled as relevant, and the closest documents are shown to the user.

Classification

A binary classifier is trained with the labels the user has given. We use a SVM with a Gaussian χ 2 kernel, as it has revealed being the most efficient [START_REF] Gosselin | Feature based approach to semi-supervised similarity learning[END_REF]. The result is a function fA y (xi) which returns the relevance of each document xi, according to the examples Ay.

Active selection

We add an active correction to the boundary (border between relevant and irrelevant) in order to deal with the few training data and the imbalance of the classes. Since our aim here is to enhance the selection, we propose to consider the boundary as the area of uncertainty, because of the interest in our context of uncertainty-based active learners in terms of efficiency and complexity. The system selects a set of documents the user should label. The selection may be such as the labeling of those documents will enhance the most the classification. We divide the selection in three steps. First, we reduce the computational time, by pre-selecting around five hundreds documents which may be in the optimal selection set (closest documents to the corrected boundary). Then we compute the selection criteria (maximizing the Average Precision) for each pre-selected documents. Finally, we compute the batch selection using the criteria.

Feedback

The user labels the selected documents, and a new classification and correction can be computed. The process is repeated as many times as necessary.

Supported media

RETIN was initially developed for CBIR context, but other medias have been integrated into the system. Videos, since it can be seen as sequence of images (video is divided into shots, key-frames are extracted for each shot and CBIR method are used for retrieval). 3D-objects retrieval is also integrated, the features extracted from the 3D-models are based on shape.

Indexing

Each document of the database is represented by features which aim at giving a complete and compact description of the content of the document, easy to compute and to deal with. These features can be used in the RETIN search engine since they are represented by a vector of the same size for one feature for each document of the database.

Images

Color and texture information are exploited. As none of color spaces has proved its superiority over the others for image coding, the HSV space is used by default. For texture analysis, Gabor filters are used with twelve different scales and orientations. Signatures are statistical distributions of colors and textures resulting from a dynamic quantization of the feature spaces. That means that we use color and texture space clustering to compute the image histograms. Both spaces are clustered using an enhanced LBG algorithm.

For a generalist database (around 10, 000 images), a small number of classes obtained by a dynamic clustering of the database is sufficient to build efficient signatures. We have adopted this dynamic quantization in RETIN with 25 classes (default value). The reader can find more details in [START_REF] Gosselin | A comparison of active classification methods for content-based image retrieval[END_REF].

3D objects

Several descriptors are available in RETIN for indexing: cord histograms, extended Gaussian images (EGI) and 3D Hough transform.

Cord histograms: A cord is defined as the vector from the object center to a vertex. Three features are defined from the cords: the length of the cord and the two angles between the cord and the first principal axis, and the second principal axis. EGI: Each 3D object is projected onto a Gaussian sphere, and each point of the sphere is valued with the area of the object faces of the same orientation. CEGI and 3D Hough are variant of EGI, all these descriptors are detailed in [START_REF] Gorisse | 3d content-based retrieval in artwork databases[END_REF].

Videos

The video indexing is performed in two parts. The first one is the segmentation of the video into shots, the second one is the computing of the features for each shot. Each shot can then be summarized by one or several key-frames.

The shot extraction is composed of two boundary detection algorithm. The first one is for detecting abrupt transitions, and the second one is for gradual transitions. To detect abrupt transitions, feature vectors with frameto-frame feature differences is computed and a kernel-based SVM classifier is used to separate cuts and non cuts after training. Once the video sequence is segmented into cut-free segments, we process a three-level gradual transition detection. The reader can find more details in [START_REF] Camara-Chavez | Robust scene cut detection by supervised learning[END_REF].

Key-frame extraction is based on a clustering of each segmented shot. For each cluster, the closest frame to the cluster center is taken as a key-frame. The features extracted for the shots are L*a*b color and Gabor texture Retin features for still images (key-frames) and the Fourier-Mellin and Zernike moments extracted for shot detection.

Graphical User Interface (GUI)

The user interface is divided in two panels (Fig. 3 (top)). On top window, documents in decreasing order of relevance are displayed, according to the current classifier. On bottom window, documents proposed for labeling are displayed, according to the current active learner. The user is invited to follow advises in bottom window, but he can choose to label documents in the top window. GUI for 3D-object and video retrieval is shown in

DEMONSTRATION

We give here an example of a RETIN search session on image database. As it is the same type of interaction for 3D and video modules, it can be generalized to all these database retrievals. In this example, the user is looking for images that represents or contains a plane.

The first step of a session is to choose an image that is belonging to the category that the user is searching. The user selects an image with a plane (the image with the green square on fig. 4).

Then, a first retrieval result is displayed (fig. 5). The classification with only one labeled image has retrieved some planes. To improve results, the user labels the 5 proposed images (all of these are relevant).

A new classification is computed with this five new labels and returns a new result (fig. 6). Then a new labeling operation is done. This time, three relevant images and two irrelevant images have been proposed.

After two other iterations, 21 images have been labelled (15 relevant, and 6 irrelevant). We must go down to the 4th page of the classified results (fig. 7) to find the first irrelevant image (all the 67 first images are relevant). This demonstration will be proposed during the conference for different media databases (3D objects, videos). FReBIR (Fuzzy Region Based Image Retrieval) is a regionbased image retrieval system, in which images are represented as adjacency graphs of fuzzy regions. The goal of this retrieval module is to retrieve images containing a specific object. An algorithm to match sub-graphs of fuzzy regions is then applied in order to retrieve images from partial queries, taking into account the image composition. Images are represented by a set of fuzzy regions, with their features (the same as global RETIN engine for images) and the composition of the image is stored into an attributed relational graph (ARG) of regions, aiming at representing the relative positions of regions. Then the problem of image retrieval from a partial query can be seen as a problem of inexact graph matching of ARGs. RETIN is used here to compute the dissimilarity between regions thanks to user labels on images. More informations about FReBIR are in [START_REF] Philipp-Foliguet | Frebir : Fuzzy region-based image retrieval[END_REF]. This module will be also available for demo during the conference.

Other modules based on RETIN

The long term learning module of RETIN uses all the labels accumulated during many interactive uses of any retrieval system to improve the feature representation of the documents. With such an optimized representation, we attempt to get a better match with semantic concepts. See [START_REF] Cord | Image retrieval using long-term semantic learning[END_REF] for more details.

In order to manage large collections of images distributed on networks, a new architecture for media retrieval in distributed media databases is currently developed, based on multi-agent systems. Our system, inspired by "ant algorithms", uses labels provided by the user for learning both the searched category of documents and the path to the most relevant databases. The problem of interactive retrieval into distributed databases context is considered with a decentralize approach. It is based on Multi-Agent Systems (MAS) offering interesting properties in comparison with centralized systems : since the agents are distributed, several sub-tasks can be processed in parallel, thus sparing both CPU and bandwidth. If a machine is down, machines still up can keep on processing the search task. More details are in [START_REF] Picard | Cbir in distributed databases using a multi-agent system[END_REF] M. Cord recently joined the LIP6 laboratory, and an extension of RETIN to text/image multimedia search engine is also currently under development.

Web interface

A web graphical interface of RETIN is already available at http://dupont.ensea.fr/~ruven/new.php. This version allows for the moment image retrieval in a generalist database. New application integration, such as 3D, videos and local search (FReBIR) is on the roadmap.

Figure 1 :

 1 Figure 1: RETIN search engine scheme.

Figure 2 :

 2 Figure 2: RETIN active learning scheme.

  Fig 3 (bottom).

Figure 3 :

 3 Figure 3: RETIN GUI: images (top), 3D (left), Videos (right).

Figure 4 :

 4 Figure 4: Initialization of the query.

Figure 5 :

 5 Figure 5: First results and first labels.

Figure 6 :

 6 Figure 6: Second results and second labels.

Figure 7 :

 7 Figure 7: Results of the 4th page.

Figure 8 :

 8 Figure 8: FReBIR: Query Regions (top), Results (bottom).

Aknowledgement

The authors are grateful to MUSCLE Network of Excellence for the financial support of this work.