No description available.

No description available.

<p>This thesis treats ultrasheaves, sheaves on the category of ultrafilters. </p><p>In the classical theory of ultrapowers, you start with an ultrafilter and, given a structure, you construct the ultrapower of the structure over the ultrafilter. The fundamental result is Los's theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure. In this thesis we instead start with the category of ultrafilters (denoted <b>U</b>). On this category <b>U</b> we build the topos of sheaves on <b>U</b> (the ultrasheaves), which we think of as generalized ultrapowers. </p><p>The theorem for ultrapowers corresponding to Los's theorem is Moerdijk's theorem, first proved by Moerdijk for the topos Sh(<b>F</b>) of sheaves on filters. In the thesis we prove that Los's theorem follows from Moerdijk's theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk's topos Sh(<b>F</b>) and prove that Sh(<b>U</b>) is the double negation subtopos of Sh(<b>F</b>). </p><p>The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category <b>U</b>. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result. </p><p>One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST) in the ultrasheaves. IST, introduced by Nelson, is a non-standard set theory, an axiomatic approach to non-standard mathematics.</p>

This thesis treats ultrasheaves, sheaves on the category of ultrafilters. In the classical theory of ultrapowers, you start with an ultrafilter and, given a structure, you construct the ultrapower of the structure over the ultrafilter. The fundamental result is Los's theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure. In this thesis we instead start with the category of ultrafilters (denoted <b>U</b>). On this category <b>U</b> we build the topos of sheaves on <b>U</b> (the ultrasheaves), which we think of as generalized ultrapowers. The theorem for ultrapowers corresponding to Los's theorem is Moerdijk's theorem, first proved by Moerdijk for the topos Sh(<b>F</b>) of sheaves on filters. In the thesis we prove that Los's theorem follows from Moerdijk's theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk's topos Sh(<b>F</b>) and prove that Sh(<b>U</b>) is the double negation subtopos of Sh(<b>F</b>). The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category <b>U</b>. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result. One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST) in the ultrasheaves. IST, introduced by Nelson, is a non-standard set theory, an axiomatic approach to non-standard mathematics.

<p>This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. </p><p>In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. </p><p>In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. </p><p>In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. </p><p>In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.</p>

This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

<p>The thesis consists of three independent parts.</p><p>Part I: Polynomial amoebas</p><p>We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1.</p><p>Part II: Differential equations in the complex plane</p><p>We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform.</p><p>Part III: Radon transforms and tomography</p><p>This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.</p>

<p>In this thesis we introduce and study a notion of effectivity (or computability) for test functions and for distributions. This is done using the theory of effective (Scott-Ershov) domains and effective domain representations.</p><p>To be able to construct effective domain representations of the spaces of test functions considered in distribution theory we need to develop the theory of admissible domain representations over countable pseudobases. This is done in the first paper of the thesis. To construct an effective domain representation of the space of distributions, we introduce and develop a notion of partial continuous function on domains. This is done in the second paper of the thesis. In the third paper we apply the results from the first two papers to develop an effective theory of distributions using effective domains. We prove that the vector space operations on each space, as well as the standard embeddings into the space of distributions effectivise. We also prove that the Fourier transform (as well as its inverse) on the space of tempered distributions is effective. Finally, we show how to use convolution to compute primitives on the space of distributions. In the last paper we investigate the effective properties of a structure theorem for the space of distributions with compact support. We show that each of the four characterisations of the class of compactly supported distributions in the structure theorem gives rise to an effective domain representation of the space. We then use effective reductions (and Turing-reductions) to study the reducibility properties of these four representations. We prove that three of the four representations are effectively equivalent, and furthermore, that all four representations are Turing-equivalent. Finally, we consider a similar structure theorem for the space of distributions supported at 0.</p>

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

In this thesis we introduce and study a notion of effectivity (or computability) for test functions and for distributions. This is done using the theory of effective (Scott-Ershov) domains and effective domain representations. To be able to construct effective domain representations of the spaces of test functions considered in distribution theory we need to develop the theory of admissible domain representations over countable pseudobases. This is done in the first paper of the thesis. To construct an effective domain representation of the space of distributions, we introduce and develop a notion of partial continuous function on domains. This is done in the second paper of the thesis. In the third paper we apply the results from the first two papers to develop an effective theory of distributions using effective domains. We prove that the vector space operations on each space, as well as the standard embeddings into the space of distributions effectivise. We also prove that the Fourier transform (as well as its inverse) on the space of tempered distributions is effective. Finally, we show how to use convolution to compute primitives on the space of distributions. In the last paper we investigate the effective properties of a structure theorem for the space of distributions with compact support. We show that each of the four characterisations of the class of compactly supported distributions in the structure theorem gives rise to an effective domain representation of the space. We then use effective reductions (and Turing-reductions) to study the reducibility properties of these four representations. We prove that three of the four representations are effectively equivalent, and furthermore, that all four representations are Turing-equivalent. Finally, we consider a similar structure theorem for the space of distributions supported at 0.