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Sjödin, Tord

Open this publication in new window or tab >>On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)### Sjödin, Tord

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_some",{id:"formSmash:j_idt191:0:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_otherAuthors",{id:"formSmash:j_idt191:0:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Probability and Mathematical Statistics, ISSN 0208-4147Article in journal (Refereed) Accepted
##### Abstract [en]

##### 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)
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Available from: 2019-11-18 Created: 2019-11-18 Last updated: 2020-06-03

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

Open this publication in new window or tab >>On the almost everywhere differentiability of the metric projection on closed sets in *l**p*(ℝ*n*), 2 < *p* < ∞### Sjödin, Tord

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_some",{id:"formSmash:j_idt191:1:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_otherAuthors",{id:"formSmash:j_idt191:1:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2018-11-04 Created: 2018-11-04 Last updated: 2018-12-18Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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

Open this publication in new window or tab >>Continuous functions and Riesz type potentials in homogeneous spaces### Sjödin, Tord

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_some",{id:"formSmash:j_idt191:2:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_otherAuthors",{id:"formSmash:j_idt191:2:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2015-06-09 Created: 2015-06-09 Last updated: 2018-06-07Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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 U^{σf}_{k} 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*.

Open this publication in new window or tab >>Explaining Kummer's test### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_some",{id:"formSmash:j_idt191:3:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_otherAuthors",{id:"formSmash:j_idt191:3:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_otherAuthors",multiple:true}); 2015 (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### 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)
#####

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Available from: 2015-06-10 Created: 2015-06-10 Last updated: 2018-06-07Bibliographically approved

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.

Open this publication in new window or tab >>On Multivariate Hyperbolically Completely Monotone Densities and Their Laplace transforms### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_some",{id:"formSmash:j_idt191:4:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_otherAuthors",{id:"formSmash:j_idt191:4:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_otherAuthors",multiple:true}); 2015 (English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### 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)
#####

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Available from: 2015-02-05 Created: 2015-02-05 Last updated: 2018-06-07Bibliographically approved

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.

Open this publication in new window or tab >>The fundamental theorem of algebra### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_some",{id:"formSmash:j_idt191:5:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_otherAuthors",{id:"formSmash:j_idt191:5:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_otherAuthors",multiple:true}); 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]

##### 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)
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##### Note

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

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 approvedOpen this publication in new window or tab >>Polar sets and capacitary potentials in homogeneous spaces### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_some",{id:"formSmash:j_idt191:6:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_otherAuthors",{id:"formSmash:j_idt191:6:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2018-06-08Bibliographically approved

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 R^{n}.

Open this publication in new window or tab >>Wolff's inequality in multi-parameter Morrey spaces### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_some",{id:"formSmash:j_idt191:7:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_otherAuthors",{id:"formSmash:j_idt191:7:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2012-07-04 Created: 2012-07-04 Last updated: 2018-06-08Bibliographically approved

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L^{p}(R^{d}) and multi-parameter Morrey spaces L^{p} λ(R^{d}), where R^{d} = R^{n1} × R^{n2} ×···× R^{nk} , λ = (λ_{1},...,λ_{k}) and 0 < λi ≤ n_{i}, 1 ≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.

Open this publication in new window or tab >>A note on Gram-Schmidt's algorithm for a general angle### Sjödin, Tord

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_some",{id:"formSmash:j_idt191:8:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_otherAuthors",{id:"formSmash:j_idt191:8:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_otherAuthors",multiple:true}); 2009 (English)In: Normat, ISSN 0801-3500, Vol. 57, no 4, p. 173-179Article in journal (Refereed) Published
##### Abstract [sv]

##### Abstract [en]

##### 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)
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##### Note

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.

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.

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 approvedOpen this publication in new window or tab >>Beurling's analyticity theorem for quantum differences### Sjödin, Tord

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##### Abstract [en]

##### 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)
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Available from: 2010-01-25 Created: 2008-06-19 Last updated: 2018-06-09Bibliographically approved

Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.

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.