| 1. |
- Uscka-Wehlou, Hanna, 1973-
(författare)
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Digital lines, Sturmian words, and continued fractions
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we present and solve selected problems arising from digital geometry and combinatorics on words. We consider digital straight lines and, equivalently, upper mechanical words with positive irrational slopes a<1 and intercept 0. We formulate a continued fraction (CF) based description of their run-hierarchical structure. Paper I gives a theoretical basis for the CF-description of digital lines. We define for each irrational positive slope less than 1 a sequence of digitization parameters which fully specifies the run-hierarchical construction. In Paper II we use the digitization parameters in order to get a description of runs using only integers. We show that the CF-elements of the slopes contain the complete information about the run-hierarchical structure of the line. The index jump function introduced by the author indicates for each positive integer k the index of the CF-element which determines the shape of the digitization runs on level k. In Paper III we present the results for upper mechanical words and compare our CF-based formula with two well-known methods, one of which was formulated by Johann III Bernoulli and proven by Markov, while the second one is known as the standard sequences method. Due to the special treatment of some CF-elements equal to 1 (essential 1's in Paper IV), our method is currently the only one which reflects the run-hierarchical structure of upper mechanical words by analogy to digital lines. In Paper IV we define two equivalence relations on the set of all digital lines with positive irrational slopes a<1. One of them groups into classes all the lines with the same run length on all digitization levels, the second one groups the lines according to the run construction in terms of long and short runs on all levels. We analyse the equivalence classes with respect to minimal and maximal elements. In Paper V we take another look at the equivalence relation defined by run construction, this time independently of the context, which makes the results more general. In Paper VI we define a run-construction encoding operator, by analogy with the well-known run-length encoding operator. We formulate and present a proof of a fixed-point theorem for Sturmian words. We show that in each equivalence class under the relation based on run length on all digitization levels (as defined in Paper IV), there exists exactly one fixed point of the run-construction encoding operator.
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| 2. |
- Andersson, Johan, 1971-
(författare)
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Summation formulae and zeta functions
- 2006
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis in analytic number theory consists of 3 parts and 13 individual papers.In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function.In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line.In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula.We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant.We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.
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| 3. |
- Jacquet, David, 1977-
(författare)
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On complex convexity
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is about complex convexity. We compare it with other notions of convexity such as ordinary convexity, linear convexity, hyperconvexity and pseudoconvexity. We also do detailed study about ℂ-convex Hartogs domains, which leads to a definition of ℂ-convex functions of class C1. The study of Hartogs domains also leads to characterization theorem of bounded ℂ-convex domains with C1 boundary that satisfies the interior ball condition. Both the method and the theorem is quite analogous with the known characterization of bounded ℂ-convex domains with C2 boundary. We also show an exhaustion theorem for bounded ℂ-convex domains with C2 boundary. This theorem is later applied, giving a generalization of a theorem of L. Lempert concerning the relation between the Carathéodory and Kobayashi metrics.
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| 4. |
- Nilsson, Lisa, 1979-
(författare)
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Amoebas, Discriminants, and Hypergeometric Functions
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of six chapters. In Chapter 1 we give some historical background to the topic of the thesis together with the fundamental definitions and results that the thesis is based on. In Chapter 2 we study Mellin transforms of rational functions and investigate their analytic continuations. The main result in this chapter is a full description of the polar locusof the meromorphic continuation of the Mellin transform. It turns out tobe closely connected with the Newton polytope of the denominator f of the rational function. We also relate the Mellin transforms to the coamoeba of the polynomial f. In fact, we represent the function 1/f as an inverse Mellin transform converging on the complement of the coamoeba of f. This is in analogy with the Laurent series expansions of 1/f which are known to converge on the complement of the amoeba of f. In Chapter 3 we study the general structure and properties of two dimensional discriminantal coamoebas. We prove that such a coamoeba is the union of two mirror images of a polygonal curve simply obtained from the matrix B in the Horn-Kapranov parametrization. We provide an area formula for the coamoeba, and show that the coamoeba is intimately related to acertain zonotope. In fact, considering the coamoeba and the zonotope as chains projected on the torus (R/2piZ)^2, the summed chain obtained as the union of the coamoeba and the zonotope is a 2-cycle, and as such, is an integer multiple of the torus itself. The last three chapters deal with hypergeometric functions, again in connection with amoeba theory. We study A-hypergeometric functions in the form of power series, and analytic continuations given by integrals of Mellin-Barnes type. We also introduce a related Gamma-integral, which is more suitable as a continuous version of the Gamma-series. We prove the orems describing the domains of convergence forA-hypergeometric series and for the associated Mellin-Barnes typeintegrals, as well as for the Gamma-integrals. The exact description of the convergence domains is given in terms of the complement components of discriminantal amoebas for the series, whereas in the case of the integrals they are given as zonotopes. By the results in Chapter 3, we know (for two dimensions) that these zonotopes exactly cover the complement of the coamoeba the correct number of times in order to get acomplete basis of hypergeometric integrals.
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| 5. |
- Sadykov, Timour, 1976-
(författare)
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Hypergeometric functions in several complex variables
- 2002
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.
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| 6. |
- Shchuplev, Alexey, 1980-
(författare)
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Toric varieties and residues
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The multidimensional residue theory as well as the theory of integral representations for holomorphic functions is a very powerful tool in complex analysis. The computation of integrals, solving algebraic or differential equations is usually reduced to some residue integral. It is a notable feature of the theory that it is based on few model differential forms. These are the Cauchy kernel and the Bochner-Martinelli kernel. These two model kernels have been the source of other fundamental kernels and residue concepts by means of homological procedures.The Cauchy and Bochner-Martinelli forms possess two common properties: firstly, their singular sets are the unions of complex subspaces, and secondly, the top cohomology group of the complement to the singular set is generated by a single element. We shall call such a set an atomic family and the corresponding form the associated residue kernel.A large class of atomic families is provided by the construction of toric varieties. The extensively developed techniques of toric geometry have already produced many explicit results in complex analysis. In the thesis, we apply these methods to the following two questions of multidimensional residue theory: simplification of the proof of the Vidras-Yger generalisation of the Jacobi residue formula in the toric setting; and construction of a residue kernel associated with a toric variety and its applications in the theory of residues and integral representations. The central role in our construction is played by the theorem stating that under some assumptions a toric variety admits realisation as a complete intersection of toric hypersurfaces in an ambient toric variety.
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| 7. |
- Rullgård, Hans, 1978-
(författare)
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Topics in geometry, analysis and inverse problems
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The thesis consists of three independent parts.Part I: Polynomial amoebasWe 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 planeWe 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 tomographyThis 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.
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