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• 101.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the almost everywhere differentiability of the metric projection on closed sets in lp(ℝn), 2 < p < ∞2018In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 68, no 143, p. 943-951Article in journal (Refereed)

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.

• 102.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Polar sets and capacitary potentials in homogeneous spaces2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, p. 771-783Article in journal (Refereed)

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.

• 103.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Thinness in non-linear potential theory for non-isotropic Sobolev spaces1997In: Ann. Acad. Aci. Fenn., Vol. 22Article in journal (Refereed)
• 104.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Weighted Lp-inequalities for multi-parameter Riesz type potentials and strong fractional maximal operators2007In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 337, p. 317-333Article in journal (Refereed)

We prove weighted L^p-inequalities  for multi-parameter Riesz type potentials, strong fractional maximal operators and their dyadic counterparts. Our proofs avoids the Good-Lambda inequalities used earlier in the R^m-case and are based on our integrated multi-parameter summation by parts lemma, that might be of independent interest.

• 105.
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics.
Weighted norm inequalities for Riesz potentials and fractional maximal functions in mixed norm Lebesgue spaces1990In: Studia Mathematica, ISSN 0039-3233, Vol. 97, no 3, p. 239-244Article in journal (Refereed)

Norm inequalities for Riesz potentials and fractional maximal functions in weighted Lebesgue spaces were proved by Muckenhoupt and Wheeden in the 1970's. We prove such inequalities in weighted mixed norm Lebesgue spaces for the full range oh indices. Our proofs make extensive use of the concept of independence of weights in the Muckenhoupt classes.

• 106.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Wolff's inequality in multi-parameter Morrey spaces2012In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 271, no 3, p. 781-787Article in journal (Refereed)

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.

• 107.
Umeå University, Faculty of Science and Technology, Department of Physics.
Determining multimediastreaming content2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This Master Thesis report was written by Umeå University Engineering Physics student Richard Tano during his thesis work at Ericsson Luleå. Monitoring network quality is of utmost importance to network providers. This can be done with models evaluating QoS (Quality of Service) and conforming to ITU-T Recommendations. When determining video stream quality there is of more importance to evaluatethe QoE (Quality of Experience) to understand how the user perceives the quality. This isranked in MOS (Mean opinion scores) values. An important aspect of determining the QoEis the video content type, which is correlated to the coding complexity and MOS values ofthe video. In this work the possibilities to improve quality estimation models complying to ITU-T study group 12 (q.14) was investigated. Methods were evaluated and an algorithm was developed that applies time series analysis of packet statistics for determination of videostreams MOS scores. Methods used in the algorithm includes a novel assembling of frequentpattern analysis and regression analysis. A model which incorporates the algorithm for usage from low to high bitrates was dened. The new model resulted in around 20% improvedprecision in MOS score estimation compared to the existing reference model. Furthermore an algorithm using only regression statistics and modeling of related statistical parameters was developed. Improvements in coding estimation was comparable with earlier algorithm but efficiency increased considerably.

• 108.
Umeå University, Faculty of Science and Technology, Department of Physics.
Source Term Estimation in the Atmospheric Boundary Layer: Using the adjoint of the Reynolds Averaged Scalar Transport equation2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This work evaluates whether the branch of Reynolds Averaging in Computational Fluid Dynamics can be used to, based on real field measurements, find the source of the measured gas in question. The method to do this is via the adjoint to the Reynolds Averaged Scalar Transport equation, explained and derived herein. Since the Inverse is only as good as the main equation, forward runs are made to evaluate the turbulence model.

Reynolds Averaged Navier Stokes is solved in a domain containing 4 cubes in a 2x2 grid, generating a velocity field for said domain. The turbulence model in question is a union of two modifications to the standard two equation k-ε model in order to capture blunt body turbulence but also to model the atmospheric boundary layer. This field is then inserted into the Reynolds Averaged Scalar Transport equation and the simulation is compared to data from the Environmental Flow wind tunnel in Surrey. Finally the adjoint scalar transport is solved, both for synthetic data that was generated in the forward run, but also for the data from EnFlo.

It was discovered that the turbulent Schmidt number plays a major role in capturing the dispersed gas, three different Schmidt numbers were tested, the standard 0.7, the unconventional 0.3 and a height dependent Schmidt number. The widely accepted value of 0.7 did not capture the dispersion at all and gave a huge model error. As such the adjoint scalar transport was solved for 0.3 and a height dependent Schmidt number.

The interaction between measurements, the real source strength (which is not used in the adjoint equation, but needed to find the source) and the location of the source is intricate indeed. Over estimation and under estimation of the forward model may cancel out in order to find the correct source, with the correct strength. It is found that Reynolds Averaged Computational fluid dynamics may prove useful in source term estimation.

• 109.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Enclosure of the Numerical Range and Resolvent Estimates of Non-Selfadjoint Operator FunctionsManuscript (preprint) (Other academic)
• 110.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Linear Functional Equations and Convergence of Iterates2012Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

The subject of this work is functional equations with direction towards linear functional equations. The .rst part describes function sets where iterates of the functions converge to a .xed point. In the second part the convergence property is used to provide solutions to linear functional equations by de.ning solutions as in.nite sums. Furthermore, this work contains some transforms to linear form, examples of functions that belong to di¤erent classes and corresponding linear functional equations. We use Mathematica to generate solutions and solve itera- tively equations.

• 111.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Spectral properties of linear operators and operator functions can be used to analyze models in nature. When dispersion and damping are taken into account, the dependence of the spectral parameter is in general non-linear and the operators are not selfadjoint.

In this thesis non-selfadjoint operator functions are studied and several methods for obtaining properties of unbounded non-selfadjoint operator functions are presented. Equivalence is used to characterize operator functions since two equivalent operators share many significant characteristics such as the spectrum and closeness. Methods of linearization and other types of equivalences are presented for a class of unbounded operator matrix functions.

To study properties of the spectrum for non-selfadjoint operator functions, the numerical range is a powerful tool. The thesis introduces an optimal enclosure of the numerical range of a class of unbounded operator functions. The new enclosure can be computed explicitly, and it is investigated in detail. Many properties of the numerical range such as the number of components can be deduced from the enclosure. Furthermore, it is utilized to prove the existence of an infinite number of eigenvalues accumulating to specific points in the complex plane. Among the results are proofs of accumulation of eigenvalues to the singularities of a class of unbounded rational operator functions. The enclosure of the numerical range is also used to find optimal and computable estimates of the norm of resolvent and a corresponding enclosure of the ε-pseudospectrum.

• 112. Vary, J. P.
Umeå University, Faculty of Science and Technology, Department of Physics.
Basis light-front quantization: a new approach to non-perturbative scattering and time-dependent production processes2013In: Acta Physica Polonica B, Proceedings Supplement, ISSN 1899-2358, E-ISSN 2082-7865, Vol. 6, no 1, p. 257-262Article in journal (Refereed)

Hamiltonian light-front quantum field theory constitutes a framework for deriving invariant masses, correlated parton amplitudes of self-bound systems and time-dependent scattering amplitudes. By choosing the lightfront gauge and adopting an orthonormal basis function representation, we obtain a large, sparse, Hamiltonian matrix for mass eigenstates that is solvable by adapting ab initio no-core methods of nuclear many-body theory. In the continuum limit, the infinite matrix limit, we recover full covariance. There is considerable freedom in the choice of the orthonormal and complete set of basis functions with key considerations being convenience and convergence properties. We adopt a two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall anti-de Sitter/quantum chromodynamics (AdS/QCD) model obtained from light-front holography. We outline our approach and present preliminary results for non-linear Compton scattering, evaluated non-perturbatively, where a strong (possibly time-dependent) laser field excites an electron that emits a photon.

• 113.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Implicit Function Theorem2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

In this essay we present an introduction to real analysis, with the purpose of proving the Implicit Function Theorem. Our proof relies on other well-known theorems in set theory and real analysis as the Heine-Borel Covering Theorem and the Inverse Function Theorem.

• 114.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Great Picard Theorem2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

In this essay, we present a proof of the great Picard theorem by showing that a holomorphic function with an essential singularity attains infinitely many complex values in the vicinity of the singularity.

• 115.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Stability patterns for a size-structured population model and its stage-structured counterpart2015In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 267, p. 109-123Article in journal (Refereed)

In this paper we compare a general size-structured population model, where a size-structured consumer feeds upon an unstructured resource, to its simplified stage-structured counterpart in terms of equilibrium stability. Stability of the size-structured model is understood in terms of an equivalent delayed system consisting of a renewal equation for the consumer population birth rate and a delayed differential equation for the resource. Results show that the size- and stage-structured models differ considerably with respect to equilibrium stability, although the two models have completely identical equilibrium solutions. First, when adult consumers are superior foragers to juveniles, the size-structured model is more stable than the stagestructured model while the opposite occurs when juveniles are the superior foragers. Second, relatively large juvenile (adult) mortality tends to stabilise (destabilise) the size-structured model but destabilise (stabilise) the stage-structured model. Third, the stability pattern is sensitive to the adult-offspring size ratio in the sizestructured model but much less sensitive in the stage-structured model. Finally, unless the adult-offspring size ratio is sufficiently small, the stage-structured model cannot satisfactorily capture the dynamics of the size-structured model. We conclude that caution must be taken when the stage-structured population model is applied, although it can consistently translate individual life history and stage-specific differences to the population level.

• 116.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Vector spaces of delta-plurisubharmonic functions and extensions of the complex Monge-Ampere operator2015In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 422, no 2, p. 960-980Article in journal (Refereed)

In this paper we shall consider two types of vector ordering on the vector space of differences of negative plurisubharmonic functions, and the problem whether it is possible to construct supremum and infimum. Then we consider two different approaches to define the complex Monge-Ampere operator on these vector spaces, and we solve some Dirichlet problems. We end this paper by stating and discussing some open problems.

• 117.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Monge-Ampère measures on subvarieties2015In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 423, p. 94-105Article in journal (Refereed)

In this article we address the question whether the complex Monge-Ampere equation is solvable for measures with large singular part. We prove that under some conditions there is no solution when the right-hand side is carried by a smooth subvariety in C-n of dimension k < n.

• 118.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Kolodziej's subsolution theorem for unbounded pseudoconvex domains2012In: Universitatis Iagellonicae Acta Mathematica, ISSN 0083-4386, 2084-3828, Vol. 50, p. 7-23Article in journal (Refereed)

In this paper we generalize Kolodziej's subsolution theorem to bounded and unbounded pseudoconvex domains, and in that way we are able to solve complex Monge-Ampère equations on general pseudoconvex domains. We then give a negative answer to a question of Cegrell and Kolodziej by constructing a compactly supported Radon measure µ that vanishes on all pluripolar sets in Cn such that µ(Cn) = (2π)n, and forwhich there is no function u in L+ such that (ddcu)=µ. We end this paper by solving a Monge-Ampère type equation. Furthermore, we proveuniqueness and stability of the solution.

• 119.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Blocki-Zwonek conjectures2015In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 60, no 9, p. 1270-1276Article in journal (Refereed)

Let Omega be a bounded pseudoconvex domain in C-n, and let g(Omega) (z, a) be the pluricomplex Green function with pole at a in Omega. It was conjectured by Blocki and Zwonek that the function given by beta = beta(Omega),(a) : (-infinity, 0) (sic) t -> beta(t) = log (lambda(n)({z is an element of Omega g(Omega) (z, a) < t})) is convex. Here.n is the Lebesgue measure in Cn. In this note we give an affirmative answer to this conjecture when Omega is biholomorphic to the unit ball or to the polydisc in C-n, n >= 1.

• 120.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Geometry of m-Hyperconvex Domains2018In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 28, no 4, p. 3196-3222Article in journal (Refereed)

We study the geometry of m-regular domains within the Caffarelli–Nirenberg–Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

• 121.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A counterexample to a conjecture by Blocki-Zwonek2018In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 27, no 1, p. 119-124Article in journal (Refereed)
• 122.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Radially symmetric plurisubharmonic functions2012In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 106, p. 1-17Article in journal (Refereed)

In this note we consider radially symmetric plurisubharmonic functions and the complex Monge–Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge–Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge–Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally we characterize radially symmetric plurisubharmonic functions among the subharmonic ones using merely the laplacian.

• 123.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Blocki-Zwonek conjectures and beyond2015In: Archiv der Mathematik, ISSN 0003-889X, E-ISSN 1420-8938, Vol. 105, no 4, p. 371-380Article in journal (Refereed)

Let $${\Omega}$$ be a bounded pseudoconvex domain in $${\mathbb{C}^n}$$, and let $${g_{\Omega}(z,a)}$$ be the pluricomplex Green function with pole at a in $${\Omega}$$. Błocki and Zwonek conjectured that the function given by $$\begin{array}{ll}\alpha = \alpha_{\Omega}, a: (- \infty, 0) \ni t \mapsto \alpha (t) = e^{-2nt} \lambda_n \left( \{z \in \Omega: g_{\Omega}(z, a) < t \} \right)\end{array}$$ is nondecreasing, and that the function given by $$\begin{array}{ll}\beta = \beta_{\Omega}, a: (-\infty, 0) \ni t \to \beta(t)= \log \left(\lambda_n \left(\{z \in \Omega: g_{\Omega}(z,a)< t\}\right)\right)\end{array}$$ is convex. Here $${\lambda_{n}}$$ is the Lebesgue measure in $${\mathbb{C}^n}$$. In this note we give an affirmative answer to these conjectures when $${\Omega}$$ is biholomorphic to a bounded, balanced, and pseudoconvex domain in $${\mathbb{C}^n}$$, $${n\geq 1}$$. The aim of this note is to consider generalizations of the functions $${\alpha}$$, $${\beta}$$ defined by the Green function with two poles in $${\mathbb{D}\subset\mathbb{C}}$$. We prove that $${\alpha}$$ is not nondecreasing, and $${\beta}$$ is not convex. By using the product property for pluricomplex Green functions, we then generalize this to n-dimensions. Finally, we end this note by considering two other possibilities generalizing the Błocki–Zwonek conjectures.

• 124.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Moser-Trudinger inequality in complex space2019In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 479, no 2, p. 1456-1474Article in journal (Refereed)

In this paper we prove the pluricomplex counterpart of the Moser-Trudinger and Sobolev inequalities in complex space. We consider these inequalities for plurisubharmonic functions with finite pluricomplex energy, and we estimate the concerned constants.

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