| 1. |
- Berkesch, Christine, et al.
(author)
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Euler-Mellin Integrals and A-Hypergeometric Functions
- 2014
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In: The Michigan mathematical journal. - : Michigan Mathematical Journal. - 0026-2285 .- 1945-2365. ; 63:1, s. 101-123
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Journal article (peer-reviewed)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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| 2. |
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| 3. |
- Nilsson, Lisa, 1979, et al.
(author)
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Mellin Transforms of Multivariate Rational Functions
- 2013
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 23:1, s. 24-46
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Journal article (peer-reviewed)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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| 4. |
- Passare, Mikael, et al.
(author)
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Amoebas of Complex Hypersurfaces in Statistical Thermodynamics
- 2013
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In: Mathematical physics, analysis and geometry. - : Springer Science and Business Media LLC. - 1385-0172 .- 1572-9656. ; 16:1, s. 89-108
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Journal article (peer-reviewed)abstract
- The amoeba of a complex hypersurface is its image under the logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble. The spectrum of the ensemble is assumed to be multidimensional; this leads us to the notions of multidimensional temperature and a vector of differential thermodynamic forms. Strictly speaking, in the paper we develop the multidimensional Darwin-Fowler method and give the description of the domain of admissible average values of energy for which the thermodynamic limit exists.
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| 5. |
- Passare, Mikael, et al.
(author)
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DISCRIMINANT COAMOEBAS THROUGH HOMOLOGY
- 2013
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In: Journal of Commutative Algebra. - 1939-0807 .- 1939-2346. ; 5:3, s. 413-440
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Journal article (peer-reviewed)abstract
- Understanding the complement of the coamoeba of a (reduced) A-discriminant is one approach to studying the monodromy of solutions to the corresponding system of A-hypergeometric differential equations. Nilsson and Passare described the structure of the coamoeba and its complement (a zonotope) when the reduced A-discriminant is a function of two variables. Their main result was that the coamoeba and zonotope form a cycle which is equal to the fundamental cycle of the torus, multiplied by the normalized volume of the set A of integer vectors. That proof only worked in dimension two. Here, we use simple ideas from topology to give a new proof of this result in dimension two, one which can be generalized to all dimensions.
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| 6. |
- Passare, Mikael, et al.
(author)
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NEW MULTIPLIER SEQUENCES VIA DISCRIMINANT AMOEBAE
- 2011
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In: Moscow Mathematical Journal. - 1609-3321 .- 1609-4514. ; 11:3, s. 547-560
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Journal article (peer-reviewed)abstract
- In their classic 1914 paper, Polya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences.
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