Browsing by Author "Mikkola, Petrus"
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Mikkola, Petrus (2020)This thesis examines discrete complex analysis and potential theory on isoradial graphs. Isoradial graphs form a general class of graphs where all faces of the graph can be inscribed into circles of equal radii. For instance, the square, the honeycomb, and the triangular lattices belong to this family. Discrete analogues (on isoradial graphs) of the classical complex analysis objects such as holomorphic and harmonic functions are considered. The focus is on two fundamental operators: the discrete Cauchy-Riemann and the discrete Laplace operator. Their inverses are studied as well: the discrete Cauchy kernel and the discrete Green’s function. The latter part of the thesis deals with discrete multiplicatively multivalued functions such as discrete complex power functions. Discrete multivalued functions are not extensively studied in general, but instead from a viewpoint of two special functions: the discrete multivalued Cauchy kernel and the discrete multivalued Green's function. These functions have relevance, for instance, when studying the asymptotics of the electric correlators of the dimer model. The dimer model is a classical model of statistical mechanics. The thesis is based on the following articles: "Discrete complex analysis on isoradial graphs" by Chelkak and Smirnov (2011), "Dimers and families of Cauchy-Riemann operators" by Dubédat (2015), and "The Laplacian and Dirac operators on critical planar graphs" by Kenyon (2002). The latter part of the thesis that deals with discrete multivalued functions, is built upon the Dubédat’s work (2015).
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Mikkola, Petrus (2018)This thesis examines the effect of investment-specific technological change on the capital replacement decision and depreciation by extending Mukoyama’s (2008) study on endogenous depreciation. When allowing investment-specific technological progress to be described either as a fall in the price of capital or as a growth in the relative productivity of new capital, and capital stock to be determined by the producer’s optimization, there arise a method to describe obsolescence as a part of depreciation and capital evolution. The following three key results are shown when assuming that scrapped capital stock has no value. First, the optimal replacement policy is stationary. Second, the acceleration of investment-specific technological progress accelerates capital replacement, hence also obsolescence. Third, whether investment-specific technological progress is modelled as a fall in the price of capital or as a growth in the relative productivity of new capital, does not impact on the optimal replacement policy. A quantitative exercise shows that the first two results seems to hold even if the scrapped capital stock has some positive value. However, if scrapped capital has some value, then the two approaches to model investment-specific technological progress are no longer equivalent. The adoption of the capital replacement problem for describing depreciation is a promising approach. Even though there does not exist a closed-from solution for the optimal replacement interval, it can be solved (in the stationary case) as a root of a relative simple transcendental function. The rate of depreciation can be explicitly solved, also in the case of non-stationary replacement policy, but that is computationally more difficult. Physical depreciation (wear and tear) can be disentangled from obsolescence insofar as either one is known. Thus, the results still rely on the estimate of physical depreciation.
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