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Sjödin, Tord
Publications (10 of 28) Show all publications
Sjödin, T. (2019). On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC). Probability and Mathematical Statistics
Open this publication in new window or tab >>On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)
2019 (English)In: Probability and Mathematical Statistics, ISSN 0208-4147Article in journal (Refereed) Accepted
Abstract [en]

Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y ·X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

Place, publisher, year, edition, pages
University of Wroclaw, Poland, 2019
Keywords
Gamma distribution, hyperbolically monotone function, Laplace transform, generalized gamma convolution (GGC)
National Category
Natural Sciences
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-165255 (URN)ISSN 0208-4147 (ISRN)10.19195/0208-4147.0.0.0 (DOI)
Available from: 2019-11-18 Created: 2019-11-18 Last updated: 2019-11-19
diva2:1260615
Open this publication in new window or tab >>On the almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞
2018 (English)In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 68, no 143, p. 943-951Article in journal (Refereed) Published
Abstract [en]

Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi(x) = xi.

Place, publisher, year, edition, pages
Berlin/Heidelberg: Springer, 2018
Keywords
normed space, uniform convexity, closed set, metric projection, l(p)-space, Frechet differential, Lipschitz condition
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-153050 (URN)10.21136/CMJ.2018.0038-17 (DOI)000451778500004 ()
Available from: 2018-11-04 Created: 2018-11-04 Last updated: 2018-12-18Bibliographically approved
Sjödin, T. (2015). Continuous functions and Riesz type potentials in homogeneous spaces. Potential Analysis, 43(3), 495-511
Open this publication in new window or tab >>Continuous functions and Riesz type potentials in homogeneous spaces
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 495-511Article in journal (Refereed) Published
Abstract [en]

We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential Uσfk of an absolutely continuous measure σ f supported in an arbitrarily small neighbourhood of K. The measure σ f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.

Place, publisher, year, edition, pages
New York: Springer-Verlag New York, 2015
Keywords
Homogeneous space, doubling measure, kernel, potential, energy, capacity, dyadic cubes
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-104307 (URN)10.1007/s11118-015-9483-4 (DOI)000365769100008 ()
Available from: 2015-06-09 Created: 2015-06-09 Last updated: 2018-06-07Bibliographically approved
Sjödin, T. (2015). Explaining Kummer's test.
Open this publication in new window or tab >>Explaining Kummer's test
2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

Mathematical theorems and their proofs are often described as being beautyful, natural or explanatory. This paper treats the last case  and discusses explanatory versus non explanatory proofs using  a convergence theorem for positive series due to Kummer, called Kummer's test. This example was studied by Pringsheim already in the beginning in the last century and has recently been discussed by Hafner and Mancuso in relation to Steiner's criteria for an explanatory proof. We provide a mathematical analysis of Kummer's test and give a proof that we claim is explanatory in this sense. Besides this we answer a number of questions that are frequently asked about the test.

Keywords
Positive series, convergence, convergence test, Kummer's test, explanatory proof
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-104407 (URN)
Available from: 2015-06-10 Created: 2015-06-10 Last updated: 2018-06-07Bibliographically approved
Sjödin, T. (2015). On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace transforms.
Open this publication in new window or tab >>On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace transforms
2015 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We study classes of multivariate (bivariate) random variables with hyperbolically completely monotone densities (MVHCM) and prove that these classes are closed with respect to the Laplace transform. We then use this result to define new classes of bivariate generalized gamma convolutions (BVGGC_L) that contain Bondesson's class BVGGC in the strong sense.

Keywords
Random vector, multivariate density, independence, Laplace transform, hyperbolocally completely monotone, generalized gamma convolution (GGC), bivariate GGC
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-99257 (URN)
Available from: 2015-02-05 Created: 2015-02-05 Last updated: 2018-06-07Bibliographically approved
Sjödin, T. (2014). The fundamental theorem of algebra (5ed.). In: Martin Aigner, Günter M. Ziegler (Ed.), Proofs from THE BOOK: (pp. 147-149). Berlin-Heidelberg: Springer
Open this publication in new window or tab >>The fundamental theorem of algebra
2014 (English)In: Proofs from THE BOOK / [ed] Martin Aigner, Günter M. Ziegler, Berlin-Heidelberg: Springer, 2014, 5, p. 147-149Chapter in book (Refereed)
Abstract [en]

Everyn nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers

Place, publisher, year, edition, pages
Berlin-Heidelberg: Springer, 2014 Edition: 5
Keywords
Polynomial, complex roots
National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-95319 (URN)10.1007/978-3-662-44205-0 (DOI)978-3-662-44204-3 (ISBN)978-3-662-44205-0 (ISBN)
Note

The BOOK, invented by the famous hungarian mathetician Paul Erdös, is supposed to be the place where God maintains the perfect proof for mathematical theorems.

Available from: 2014-10-27 Created: 2014-10-27 Last updated: 2018-06-07Bibliographically approved
Sjödin, T. (2013). Polar sets and capacitary potentials in homogeneous spaces. Annales Academiae Scientiarum Fennicae Mathematica, 38, 771-783
Open this publication in new window or tab >>Polar sets and capacitary potentials in homogeneous spaces
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, p. 771-783Article in journal (Refereed) Published
Abstract [en]

A set E in a space X is called a polar set in X, relative to a kernel k(x; y), if thereis a nonnegative measure in X such that the potential Uk(x) = ∞ precisely when x ∈ E. Polarsets have been characterized in various classical cases as G-sets (countable intersections of opensets) with capacity zero. We characterize polar sets in a homogeneous space (X; d; ) for severalclasses of kernels k(x; y), among them the Riesz -kernels and logarithmic Riesz kernels. The latercase seems to be new even in Rn.

Place, publisher, year, edition, pages
Helsingfors: , 2013
Keywords
Metric space, doubling measure, definite kernel, consistent kernel, potential, energy, capacity, polar set
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-79308 (URN)10.5186/aasfm.2013.3851 (DOI)000322091900023 ()
Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2018-06-08Bibliographically approved
Sjödin, T. (2012). Wolff's inequality in multi-parameter Morrey spaces. Mathematische Zeitschrift, 271(3), 781-787
Open this publication in new window or tab >>Wolff's inequality in multi-parameter Morrey spaces
2012 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 271, no 3, p. 781-787Article in journal (Refereed) Published
Abstract [en]

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces Lp(Rd) and multi-parameter Morrey spaces Lp λ(Rd), where Rd = Rn1 × Rn2 ×···× Rnk , λ = (λ1,...,λk) and 0 < λi ≤ ni, 1 ≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.

Place, publisher, year, edition, pages
Berlin: Springer, 2012
Keywords
Riesz potential, Wolff potential, multi-parameter potential, dyadic potential, Wolff inequality, Morrey space
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-57086 (URN)10.1007/s00209-011-0890-y (DOI)000306342700011 ()
Available from: 2012-07-04 Created: 2012-07-04 Last updated: 2018-06-08Bibliographically approved
Sjödin, T. (2009). A note on Gram-Schmidt's algorithm for a general angle. Normat, 57(4), 173-179
Open this publication in new window or tab >>A note on Gram-Schmidt's algorithm for a general angle
2009 (English)In: Normat, ISSN 0801-3500, Vol. 57, no 4, p. 173-179Article in journal (Refereed) Published
Abstract [sv]

Gram-Schmidts algoritm ger en parvis ortogonal mängd från en linjärt oberoende mängd i ett allmännt vektorrum med inre produkt. Vi ger en algoritm som konstruerar en mängd där vektorerna parvis har en godtycklig på förhand given vinkel, i alla fall där detta är möjligt.

Abstract [en]

The Gram-Schmidt algorithm produces a pairwise ortogonal set of vectors from a linearly independent set in a general vector space with an inner product. We give an algorithm that produces vectors that have pairwise the same angle, in all cases where this is possible.

Place, publisher, year, edition, pages
Göteborg: , 2009
Keywords
inner product, Gram-Schmidt, arbitrary angel, Inre product, Gram-Schmidt, godtycklig vinkel
National Category
Engineering and Technology Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-54209 (URN)
Note

Svensk alternativ titel: "Gram-Schmidt´s algoritm för en allmän vinkel".

Available from: 2012-04-19 Created: 2012-04-19 Last updated: 2018-06-08Bibliographically approved
Sjödin, T. (2008). Beurling's analyticity theorem for quantum differences. Manuscripta mathematica, 127(3), 369-380
Open this publication in new window or tab >>Beurling's analyticity theorem for quantum differences
2008 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 127, no 3, p. 369-380Article in journal (Refereed) Published
Abstract [en]

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.

Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2008
Keywords
quantum difference, quantum analysis, analyticity
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-10136 (URN)10.1007/s00229-008-0213-8 (DOI)
Available from: 2010-01-25 Created: 2008-06-19 Last updated: 2018-06-09Bibliographically approved
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