| 1. |
- 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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| 2. |
- Berkesch, Christine, et al.
(författare)
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Euler-Mellin Integrals and A-Hypergeometric Functions
- 2014
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Ingår i: The Michigan mathematical journal. - : Michigan Mathematical Journal. - 0026-2285 .- 1945-2365. ; 63:1, s. 101-123
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Tidskriftsartikel (refereegranskat)abstract
- We consider integrals that generalize both Mellin transforms of rational functions of the form 1/f and classical Euler integrals. The domains of integration of our so-called Euler Mellin integrals are naturally related to the coamoeba of f, and the components of the complement of the closure of this coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin Barnes integrals.
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| 3. |
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| 4. |
- 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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| 5. |
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| 6. |
- Melin, Erik, 1980-
(författare)
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Digital Geometry and Khalimsky Spaces
- 2008
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Digital geometry is the geometry of digital images. Compared to Euclid’s geometry, which has been studied for more than two thousand years, this field is very young.Efim Khalimsky’s topology on the integers, invented in the 1970s, is a digital counterpart of the Euclidean topology on the real line. The Khalimsky topology became widely known to researchers in digital geometry and computer imagery during the early 1990s.Suppose that a continuous function is defined on a subspace of an n-dimensional Khalimsky space. One question to ask is whether this function can be extended to a continuous function defined on the whole space. We solve this problem. A related problem is to characterize the subspaces on which every continuous function can be extended. Also this problem is solved.We generalize and solve the extension problem for integer-valued, Khalimsky-continuous functions defined on arbitrary smallest-neighborhood spaces, also called Alexandrov spaces.The notion of a digital straight line was clarified in 1974 by Azriel Rosenfeld. We introduce another type of digital straight line, a line that respects the Khalimsky topology in the sense that a line is a topological embedding of the Khalimsky line into the Khalimsky plane.In higher dimensions, we generalize this construction to digital Khalimsky hyperplanes, surfaces and curves by digitization of real objects. In particular we study approximation properties and topological separation properties. The last paper is about Khalimsky manifolds, spaces that are locally homeomorphic to n-dimensional Khalimsky space. We study different definitions and address basic questions such as uniqueness of dimension and existence of certain manifolds.
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| 7. |
- 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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| 8. |
- Nilsson, Lisa, et al.
(författare)
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Discriminant coamoebas in dimension two
- 2010
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Ingår i: Journal of Commutative Algebra. - 1939-0807 .- 1939-2346. ; 2:4, s. 447-471
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Tidskriftsartikel (refereegranskat)
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| 9. |
- Nilsson, Lisa, et al.
(författare)
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Domains of Convergence for A-hypergeometric Series and Integrals
- 2019
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Ingår i: Journal of Siberian Federal University Mathematics and Physics. - : Siberian Federal University. - 1997-1397 .- 2313-6022. ; 12:4, s. 509-529
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Tidskriftsartikel (refereegranskat)abstract
- We prove two theorems on the domains of convergence for A-hypergeometric series and for associated Mellin-Barnes type integrals. The exact convergence domains are described in terms of amoebas and coamoebas of the corresponding principal A-determinants.
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| 10. |
- Nilsson, Lisa, 1979, et al.
(författare)
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Mellin Transforms of Multivariate Rational Functions
- 2013
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 23:1, s. 24-46
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Tidskriftsartikel (refereegranskat)abstract
- This paper deals with Mellin transforms of rational functions g/f in several variables. We prove that the polar set of such a Mellin transform consists of finitely many families of parallel hyperplanes, with all planes in each such family being integral translates of a specific facial hyperplane of the Newton polytope of the denominator f. The Mellin transform is naturally related to the so-called coamoeba , where Z (f) is the zero locus of f and Arg denotes the mapping that takes each coordinate to its argument. In fact, each connected component of the complement of the coamoeba gives rise to a different Mellin transform. The dependence of the Mellin transform on the coefficients of f, and the relation to the theory of A-hypergeometric functions is also discussed in the paper.
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