Let X denote a complete separable metric space, and let C(X ) denote the linear space of all bounded continuous real-valued functions on X. A semigroup T of transformations from X into X is said to be jointly continuous if the mapping (t, x) Ä T(t) x is jointly continuous from [0, )_X into X. The Lie generator of such a semigroup T is the linear operator in C(X) consisting of all ordered pairs ( f, g) such that f, g # C(X ), and for each x # X, g(x) is the derivative at 0 of f (T( } ) x). We completely characterize such Lie generators and establish the canonical exponential formula for the original semigroup in terms of powers of resolvents of its Lie generator. The only topological notions needed in the characterization are two notions of sequential convergence, pointwise and strict. A sequence in C(X) converges strictly if the sequence is uniformly bounded in the supremum norm and converges uniformly on compact subsets of X.

Learning is often perceived as a cost-reducing endogenous by-product of production processes. In many applications this by-product is modeled as a learning curve; that is, a simple function of time or of cumulative production experience. In an earlier paper we presented an alternative explanation where managers decide what resources to devote to knowledge acquisition. In this paper we expand those results to a situation using a more flexible production technology and emphasizing discounted cost. Our model explains resource and output behavior for a firm that is producing specialized units to contractual order. However, the results are quite general and have implications for investment in research, engineering, science and technology, software development, and worker training. We provide examples where the cost-minimizing producer will choose to invest in knowledge creation early in the production program and then have the rate of investment decline over time. Other interesting results are noted by examining the optimal time paths of the control and state variables in a comparative dynamic analysis.learning augmented planning models, dynamic optimization, optimal control theory, knowledge creation

A solution is developed for a convection-diffusion equation describing chemical transport with sorption, decay, and production. The problem is formulated in a finite domain where the appropriate conservation law yields Robin conditions at the ends. When the input concentration is arbitrary, the problem is underdetermined because of an unknown exit concentration. We resolve this by defining the exit concentration as a solution to a similar diffusion equation which satisfies a Dirichlet condition at the left end of the half line. This problem does not appear to have been solved in the literature, and the resulting representation should be useful for problems of practical interest.Authors of previous works on problems of this type have eliminated the unknown exit concentration by assuming a continuous concentration at the outflow boundary. This yields a wellposed problem by forcing a homogeneous Neumann exit, widely known as the Danckwerts [1] condition. We provide a solution to the Neumann problem and use it to produce an estimate which demonstrates that the Danckwerts condition implies a zero concentration at the outflow boundary, even for a long flow domain and a large time.

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SUMMARYA new finite element, viable for use in the three-dimensional simulation of transient physical processes with sharply varying solutions, is presented. The element is intended to function in adaptive h-refinement schemes as a versatile transition between regions of different refinement levels, ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries, and retaining all degrees of freedom in the solution phase. Construction of the element shape functions is described, and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem. The new element can be used in moving-front problems, such as those found in reservoir engineering and groundwater flow applications.