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• 1. Abatangelo, Laura
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Spectral stability under removal of small capacity sets and applications to Aharonov–Bohm operators2019Inngår i: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 9, nr 2, s. 379-427Artikkel i tidsskrift (Fagfellevurdert)

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov–Bohm operators with two colliding poles moving on an axis of symmetry of the domain.

• 2. Abatangelo, Laura
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications2020Inngår i: ESAIM. COCV, ISSN 1292-8119, E-ISSN 1262-3377Artikkel i tidsskrift (Fagfellevurdert)

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles.

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• 3.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Asymptotic distribution of zeros of a certain class of hypergeometric polynomials2014Licentiatavhandling, monografi (Annet vitenskapelig)

The thesis consists of two papers, both treating hypergeometric polynomials, and a short introduction. The main results are as follows.In the first paper,we study the asymptotic zero distribution of a family of hypergeometric polynomials in one complex variable as their degree goes to infinity,using the associated differential equations that hypergeometric polynomials satisfy.   We describe in particular the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation.   The new result is first of all that we are able to formulate results on the location of zeros of generalized hypergeometric polynomials in greater generality than before (earlier results are mainly concerned with the Gauss hypergeometric case.) Secondly, we are able to formulate a precise conjucture giving the asymptotic behaviour of zeros in the generalized case of our polynomials, which covers previous results.In the second paper we partly prove one of the  conjectures in the first paper by using Euler integral representation of the Gauss hypergeometric functions together with the Saddle point method.

• 4.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Addis Ababa University, Ethiopia.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomialsd2016Inngår i: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 16, nr 2, s. 167-185Artikkel i tidsskrift (Fagfellevurdert)

We study the asymptotic behavior of the zeros of a family of a certain class of hypergeometric polynomials [GRAPHICS] , using the associated hypergeometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525-568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275-287, 2001), Driver and Duren (Algorithms 21(1-4):147-156, 1999), and Duren and Guillou (J Approx Theory 111(2):329-343, 2001), and present experimental evidence to substantiate it.

• 5.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Addis Ababa University, Ethiopia.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
ZEROS OF A CERTAIN CLASS OF GAUSS HYPERGEOMETRIC POLYNOMIALS2018Inngår i: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 68, nr 4, s. 1021-1031Artikkel i tidsskrift (Fagfellevurdert)

We prove that as n -> infinity, the zeros of the polynomial F-2(1) 9-n, (an + 2) (an + 1) ; z] cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999-2001) to the case of a complex parameter alpha and partially proves a conjecture made by the authors in an earlier work.

• 6.
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
Nyanlända elevers svårigheter i algebra: En studie om nyanlända elevers uppfattningar om undervisning i algebra samt textuppgifter inom algebra i introduktionsprogram2018Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave

Syftet med denna studie är att undersöka vilka matematiska och språkliga svårigheter nyanlända gymnasieelever har när de arbetar med algebra, både deras uppfattningar om undervisning i algebra och hur de löser textuppgifter inom algebra. Samtliga elever i studien är nyanlända och går ett introduktionsprogram och de har ett annat modersmål än svenska.

Flera studier visar på att elever som har ett annat modersmålspråk än svenska har svårare att klara matematik i skolan och därmed presterar sämre i matematikundervisningen än andra elever som har svenska som modersmål (Malmer, 2002). I denna studie undersöks vad detta beror på och hur undervisningen kan anpassas för att bättre gynna den berörda elevgruppen.

För att besvara frågeställningarna gjordes elevintervjuer med åtta elever samt ett test i algebra med eleverna som deltog i intervjuerna. Resultaten i denna studie visar på att eleverna hade språkliga svårigheter som påverkade deras problemlösningsförmåga i algebra. En orsak till detta är språkliga svårigheter och detta kan delvis bero på bristfälliga svenskkunskaper som leder till svårigheter att förstå textuppgifter och svårigheter att uttrycka sig när man kommunicerar inom matematik.

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• 7.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Decomposition of D-modules over a hyperplane arrangement in the plane2010Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 48, nr 2, s. 211-229Artikkel i tidsskrift (Fagfellevurdert)

Let alpha(1), alpha(2),..., alpha(m) be linear forms defined on C-n and X = C-n\boolean OR(m)(i=1) V(alpha(i)), where V(alpha(i))={p is an element of C-n : alpha(i)(p)=0}. The coordinate ring O-X of X is a holonomic A(n)-module, where A(n) is the nth Weyl algebra and since holonomic A(n)-modules have finite length, O-X has finite length. We consider a "" twisted"" variant of this An- module which is also holonomic. Define M-alpha(beta) to be the free rank-1 C[x](alpha)-module on the generator alpha(beta) (thought of as a multivalued function), where alpha(beta)=alpha(beta 1)(1),..., alpha(beta m)(m) and the multi-index beta=(beta(1),...,beta(m))is an element of C-m. Our main result is the computation of the number of decomposition factors of M-alpha(beta) and their description when n-2.

• 8.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Addis Ababa University, Ethiopia.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Decomposition factors of D-modules on hyperplane configurations in general position2012Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, nr 8, s. 2699-2711Artikkel i tidsskrift (Fagfellevurdert)

Let alpha(1), ... , alpha(m) be linear functions on C-n and X = C-n \ V(alpha), where alpha = Pi(m)(i=1) alpha(i) and V(alpha) = {p is an element of C-n : alpha(p) = 0}. The coordinate ring O-X = C[x](alpha) of X is a holonomic A(n)-module, where A(n) is the n-th Weyl algebra, and since holonomic A(n)-modules have finite length, O-X has finite length. We consider a twisted variant of this A(n)-module which is also holonomic. Define M-alpha(beta) to be the free rank 1 C[x](alpha)-module on the generator alpha(beta) (thought of as a multivalued function), where alpha(beta) = alpha(beta 1)(1) ... alpha(beta m)(m) and the multi-index beta = (beta(1), ... , beta(m)) is an element of C-m. It is straightforward to describe the decomposition factors of M-alpha(beta), when the linear functions alpha(1), ... , alpha(m) define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on beta for the irreducibility of M-alpha(beta), in terms of numerical data for a resolution of the singularities of V(alpha).

• 9.
Stockholms universitet.
Contributions to three problems in systems of differential and convolution equations1984Doktoravhandling, med artikler (Annet vitenskapelig)
• 10. Abtahi, Yasmine
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
Otherness in mathematics education2018Inngår i: Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education / [ed] E. Bergqvist, M. Österholm, C. Granberg, L. Sumpter, Umeå, Sweden: PME , 2018, Vol. 1, s. 95-124Konferansepaper (Fagfellevurdert)
• 11. Aceto, Luca
Stockholms universitet, Humanistiska fakulteten, Filosofiska institutionen. University of Johannesburg, South Africa.
A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases2016Inngår i: Acta Informatica, ISSN 0001-5903, E-ISSN 1432-0525, Vol. 53, nr 3, s. 207-246Artikkel i tidsskrift (Fagfellevurdert)

Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

• 12. Ackermann, Nils
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
A concentration phenomenon for semilinear elliptic equations2013Inngår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 207, nr 3, s. 1075-1089Artikkel i tidsskrift (Fagfellevurdert)

For a domain $\Omega\subset\dR^N$ we consider the equation $-\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\Omega$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in\Omega$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.

Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
CUTTING DOWN TREES WITH A MARKOV CHAINSAW2014Inngår i: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 24, nr 6, s. 2297-2339Artikkel i tidsskrift (Fagfellevurdert)

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton-Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny n. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton-Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge.

Stockholms universitet, Samhällsvetenskapliga fakulteten, Statistiska institutionen.
Empirical properties of closed- and open-economy DSGE models of the Euro area2008Inngår i: Macroeconomic dynamics (Print), ISSN 1365-1005, E-ISSN 1469-8056, Vol. 12, s. 2-19Artikkel i tidsskrift (Fagfellevurdert)

In this paper, we compare the empirical proper-ties of closed- and open-economy DSGE models estimated on Euro area data. The comparison is made along several dimensions; we examine the models in terms of their marginal likelihoods, forecasting performance, variance decompositions, and their transmission mechanisms of monetary policy.

Stockholms universitet, Samhällsvetenskapliga fakulteten, Statistiska institutionen.
Forecasting performance of an open economy DSGE model2007Inngår i: Econometric Reviews, ISSN 0747-4938, E-ISSN 1532-4168, Vol. 26, nr 04-feb, s. 289-328Artikkel i tidsskrift (Fagfellevurdert)

This paper analyzes the forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model, estimated with Bayesian methods, for the Euro area during 1994Q1-2002Q4. We compare the DSGE model and a few variants of this model to various reduced form forecasting models such as vector autoregressions (VARs) and vector error correction models (VECM), estimated both by maximum likelihood and, two different Bayesian approaches, and traditional benchmark models, e.g., the random. walk. The accuracy of point forecasts, interval forecasts and the predictive distribution as a whole are assessed in, an out-of-sample rolling event evaluation using several univariate and multivariate measures. The results show that the open economy DSGE model compares well with more empirical models and thus that the tension between, rigor and fit in older generations of DSGE models is no longer present. We also critically examine the role of Bayesian model probabilities and other frequently used low-dimensional summaries, e.g., the log determinant statistic, as measures of overall forecasting performance.

• 16.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Hardy and spectral inequalities for a class of partial differential operators2014Doktoravhandling, monografi (Annet vitenskapelig)

This thesis is devoted to the study of Hardy and spectral inequalities for the Heisenberg and the Grushin operators. It consists of five chapters. In chapter 1 we present basic notions and summarize the main results of the thesis. In chapters 2-4 we deal with different types of Hardy inequalities for Laplace and Grushin operators with magnetic and non-magnetic fields. It was shown in an article by Laptev and Weidl that for some magnetic forms in two dimensions, the Hardy inequality holds in its classical form. More precisely, by considering the Aharonov-Bohm magnetic potential, we can improve the constant in the respective Hardy inequality. In chapter 2 we establish an Lp - Hardy inequality related to Laplacians with magnetic fields with Aharonov-Bohm vector potentials. In chapter 3 we introduce a suitable notion of a vector field for the Grushin sub-elliptic operator G and obtain an improvement of the Hardy inequality, which was previously obtained in the paper of N. Garofallo and E. Lanconelli. In chapter 4 we find an Lp version of the Hardy inequality obtained in chapter 2. Finally in chapter 5 we aim to find the CLR and Lieb-Thirringbninequalities for harmonic Grushin-type operators. As the Grushin operator is non-elliptic, these inequalities will not take their classical form.

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• 17.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Hardy inequalities for a magnetic Grushin operator with Aharonov-Bohm type magnetic field2012Inngår i: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 23, nr 2, s. 203-208Artikkel i tidsskrift (Fagfellevurdert)

A version of the Aharonov-Bohm magnetic field for a Grushin subelliptic operator is introduced; then its quadratic form is shown to satisfy an improved Hardy inequality.

• 18. Agbor, Dieudonné
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On the Modulus of Continuity of Mappings Between Euclidean Spaces2013Inngår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 112, nr 1, s. 147-160Artikkel i tidsskrift (Fagfellevurdert)

Let f be a function from R-P to R-q and let Lambda be a finite set of pairs (theta, eta) is an element of R-P x R-q. Assume that the real-valued function (eta, f(x)) is Lipschitz continuous in the direction theta for every (theta, eta) is an element of Lambda. Necessary and sufficient conditions on Lambda are given for this assumption to imply each of the following: (1) that f is Lipschitz continuous, and (2) that f is continuous with modulus of continuity <= C epsilon vertical bar log epsilon vertical bar.

• 19. Agmon, Shmuel
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Persistence of embedded eigenvalues2011Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 261, nr 2, s. 451-477Artikkel i tidsskrift (Fagfellevurdert)

We consider conditions under. which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < infinity we show that in favorable situations, the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of codimension in. We also have results regarding the cases when the eigenvalue is degenerate or when the multiplicity of the continuous spectrum is infinite.

• 20.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
A temporal perspective on the rate of convergence in first-passage percolation under a moment condition2019Inngår i: Brazilian Journal of Probability and Statistics, ISSN 0103-0752, E-ISSN 2317-6199, Vol. 33, nr 2, s. 397-401Artikkel i tidsskrift (Fagfellevurdert)

We study the rate of convergence in the celebrated Shape Theorem in first-passage percolation, obtaining the precise asymptotic rate of decay for the probability of linear order deviations under a moment condition. Our results are presented from a temporal perspective and complement previous work by the same author, in which the rate of convergence was studied from the standard spatial perspective.

• 21.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Noise sensitivity and Voronoi percolation2018Inngår i: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 23, artikkel-id 108Artikkel i tidsskrift (Fagfellevurdert)

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on R-2. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

• 22.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Competing first passage percolation on random graphs with finite variance degrees2019Inngår i: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, nr 3, s. 545-559Artikkel i tidsskrift (Fagfellevurdert)

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if lambda(1) = lambda(2), then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V is an element of (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If lambda(1) not equal lambda(2), on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

• 23.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Competition in growth and urns2019Inngår i: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, nr 2, s. 211-227Artikkel i tidsskrift (Fagfellevurdert)

We study survival among two competing types in two settings: a planar growth model related to two-neighbor bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or infinity, depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls ( but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one-for-one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two-type growth model on Z(2), one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

• 24.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Existence of an unbounded vacant set for subcritical continuum percolation2018Inngår i: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 23, artikkel-id 63Artikkel i tidsskrift (Fagfellevurdert)

We consider the Poisson Boolean percolation model in R-2, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in R-d, for any d >= 2, finite moment of order d is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.

• 25. Ahlkrona, Josefin
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för naturgeografi och kvartärgeologi (INK).
A numerical study of scaling relations for non-Newtonian thin film flows with applications in ice sheet modelling2013Inngår i: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 66, nr 4, s. 417-435Artikkel i tidsskrift (Fagfellevurdert)

This article treats the viscous, non-Newtonian thin-film flow of ice sheets, governed by the Stokes equations, and the modelling of ice sheets with asymptotic expansion of the analytical solutions in terms of the aspect ratio, which is a small parameter measuring the shallowness of an ice sheet. An asymptotic expansion requires scalings of the field variables with the aspect ratio. There are several, conflicting, scalings in the literature used both for deriving simplified models and for analysis. We use numerical solutions of the Stokes equations for varying aspect ratios in order to compute scaling relations. Our numerically obtained results are compared with three known theoretical scaling relations: the classical scalings behind the Shallow Ice Approximation, the scalings originally used to derive the so-called Blatter-Pattyn equations, and a non-uniform scaling which takes into account a high viscosity boundary layer close to the ice surface. We find that the latter of these theories is the most appropriate one since there is indeed a boundary layer close to the ice surface where scaling relations are different than further down in the ice. This boundary layer is thicker than anticipated and there is no distinct border with the inner layer for aspect ratios appropriate for ice sheets. This makes direct application of solutions obtained by matched asymptotic expansion problematic.

• 26. Ahlkrona, Josefin
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för naturgeografi.
Dynamically coupling the non-linear Stokes equations with the shallow ice approximation in glaciology: Description and first applications of the ISCAL method2016Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 308, s. 1-19Artikkel i tidsskrift (Fagfellevurdert)

We propose and implement a new method, called the Ice Sheet Coupled Approximation Levels (ISCAL) method, for simulation of ice sheet flow in large domains during long time-intervals. The method couples the full Stokes (FS) equations with the Shallow Ice Approximation (SIA). The part of the domain where SIA is applied is determined automatically and dynamically based on estimates of the modeling error. For a three dimensional model problem, ISCAL computes the solution substantially faster with a low reduction in accuracy compared to a monolithic FS. Furthermore, ISCAL is shown to be able to detect rapid dynamic changes in the flow. Three different error estimations are applied and compared. Finally, ISCAL is applied to the Greenland Ice Sheet on a quasi-uniform grid, proving ISCAL to be a potential valuable tool for the ice sheet modeling community.

• 27. Ahmed, S. Ejaz
Stockholms universitet, Samhällsvetenskapliga fakulteten, Statistiska institutionen.
Estimation of Several Intraclass Correlation Coefficients2015Inngår i: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 44, nr 9, s. 2315-2328Artikkel i tidsskrift (Fagfellevurdert)

An intraclass correlation coefficient observed in several populations is estimated. The basis is a variance-stabilizing transformation. It is shown that the intraclass correlation coefficient from any elliptical distribution should be transformed in the same way. Four estimators are compared. An estimator where the components in a vector consisting of the transformed intraclass correlation coefficients are estimated separately, an estimator based on a weighted average of these components, a pretest estimator where the equality of the components is tested and then the outcome of the test is used in the estimation procedure, and a James-Stein estimator which shrinks toward the mean.

• 28. Ahrens, Benedikt
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Displayed Categories2019Inngår i: Logical Methods in Computer Science, ISSN 1860-5974, E-ISSN 1860-5974, Vol. 15, nr 1, artikkel-id 20Artikkel i tidsskrift (Fagfellevurdert)

We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to 'a category D and functor F : D -> C', but instead of having a single collection of 'objects of D' with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

• 29.
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
Hur kan elevers kunskapsutveckling i matematik förbättras: Formativ bedömning i matematikundervisning2010Independent thesis Advanced level (professional degree), 10 poäng / 15 hpOppgave

Syftet med examensarbetet är att fördjupa vår kunskap om och förståelse för formativ bedömning i matematik. Vidare vill vi undersöka om den formativa bedömningen kan förbättra eleverna i klassens kunskapsutveckling i matematik. Vi har gjort en fallstudie i en klass som arbetar formativt för att undersöka hur det kan se ut i praktiken. Underlaget för det samlade materialet består av observation och intervju för att besvara våra två frågeställningar som följer. Hur kan en formativ bedömning se ut i praktiken? Hur kan en koppling mellan lärares och elevers uppfattningar om den pedagogiska verksamheten se ut? Vi har kommit fram till att den formativa bedömningen kan förbättra eleverna i klassens kunskapsutveckling i matematik. I den formativa bedömningen har vi sett vikten av mötet mellan lärare och elev. Att arbeta formativt är tidskrävande.

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Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
MIRROR SYMMETRY AND THE CLASSIFICATION OF ORBIFOLD DEL PEZZO SURFACES2016Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, nr 2, s. 513-527Artikkel i tidsskrift (Fagfellevurdert)

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.

• 31.
King Faisal University, Saudi Arabia.
Extension of Positive currents with Special Properties of Monge-Ampere Operators2013Inngår i: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 113, nr 1, s. 108-127Artikkel i tidsskrift (Fagfellevurdert)

This paper deals with the extension of positive currents across different types of sets. For closed complete pluripolar obstacles, we show the existence of such extensions. To do so, further Hausdorff dimension conditions are required. Moreover, we study the case when these obstacles are zero sets of strictly $k$-convex functions.

• 32.
The Department of Mathematics, King Faisal University.
The extendability of S-plurisubharmonic currents:  = Sur le prolongement des courants S-plurisousharmoniques2012Inngår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 350, nr 23-24, s. 1023-1026Artikkel i tidsskrift (Annet vitenskapelig)

This paper studies the extendability of negative S-plurisubharmonic current of bidimension (p,p) across a (2p-2)-dimensional closed set. Using only the positivity of S, we show that such extensions exist in the case when these obstacles are complete pluripolar, as well as zero sets of C2-plurisubharmoinc functions.

• 33.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
The inductive wedge product of positive currentsManuskript (preprint) (Annet vitenskapelig)

In this paper, we discuss the wedge product of positive pluriharmonic (resp. plurisubharmonic) current of bidimension $(p,p)$ with the Monge-Ampère operator of plurisubharmonic function. In the first part of the paper, we define this product when the locus points of the plurisubharmonic function are located in a (2p-2)-dimensional closed set (resp. (2p-4)-dimensional sets), in the sense of Hartogs. The second part treats the case when these locus points are contained in a compact complete pluripolar sets and p≥ 2 (resp. p≥3).

• 34.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On the Extension and Wedge Product of Positive Currents2012Doktoravhandling, med artikler (Annet vitenskapelig)

This dissertation is concerned with extensions and wedge products of positive currents. Our study can be considered as a generalization for classical works done earlier in this field.

Paper I deals with the extension of positive currents across different types of sets. For closed complete pluripolar obstacles, we show the existence of such extensions. To do so, further Hausdorff dimension conditions are required. Moreover, we study the case when these obstacles are zero sets of strictly k-convex functions.

In Paper II, we discuss the wedge product of positive pluriharmonic (resp. plurisubharmonic) current of bidimension (p,p) with the Monge-Ampère operator of plurisubharmonic function. In the first part of the paper, we define this product when the locus points of the plurisubharmonic function are located in a (2p-2)-dimensional closed set (resp. (2p-4)-dimensional sets), in the sense of Hartogs. The second part treats the case when these locus points are contained in a compact complete pluripolar sets and p≥2 (resp. p≥3).

Paper III studies the extendability of negative S-plurisubharmonic current of bidimension (p,p) across a (2p-2)-dimensional closed set. Using only the positivity of S, we show that such extensions exist in the case when these obstacles are complete pluripolar, as well as zero sets of C2-plurisubharmoinc functions.

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• 35. Albert, Michael
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Embedding Small Digraphs and Permutations in Binary Trees and Split Trees2020Inngår i: Algorithmica, ISSN 0178-4617, E-ISSN 1432-0541, Vol. 82, nr 3, s. 589-615Artikkel i tidsskrift (Fagfellevurdert)

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335-364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409-432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

• 36. Albeverio, Sergio
Elander, NilsStockholms universitet, Naturvetenskapliga fakulteten, Fysikum.
Operator methods in ordinary and partial differential equations: proceedings of the Sonja Kovalevsky symposium held in the University of Stockholm, June 16-22, 20002002Konferanseproceedings (Fagfellevurdert)
• 37. Albeverio, Sergio
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Singular perturbations of differential operators: solvable Schrödinger type operators2000Bok (Fagfellevurdert)
• 38.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Combinatorial Methods in Complex Analysis2013Doktoravhandling, med artikler (Annet vitenskapelig)

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts.

Part A: Spectral properties of the Schrödinger equation

This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained.

Part B: Graph monomials and sums of squares

In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares.

Part C: Eigenvalue asymptotics of banded Toeplitz matrices

This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above.

Part D: Stretched Schur polynomials

This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients.

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• 39.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
LLT polynomials, elementary symmetric functions and melting lollipops2020Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Artikkel i tidsskrift (Fagfellevurdert)

We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary symmetric basis for the class of graphs called melting lollipops previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs. We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials, and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

• 40.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential2012Inngår i: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 12, nr 2, s. 465-481Artikkel i tidsskrift (Fagfellevurdert)

In this paper, we generalize several results in the article Analytic continuation of eigenvalues of a quartic oscillator of A. Eremenko and A. Gabrielov [4]. We consider a family of eigenvalue problems for a Schrodinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k < (d + 2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k = (d + 2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components.

• 41.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential2010Licentiatavhandling, med artikler (Annet vitenskapelig)

In this thesis, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schroedinger equation with quartic potentials.

In the first paper, we consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.

In the second paper, we only consider even polynomial potentials, and show that the spectral determinant for the eigenvalue problem consists of two irreducible components. A similar result to that of paper I is proved for k boundary conditions.

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• 42.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Schur polynomials, banded Toeplitz matrices and Widom's formula2012Inngår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, nr 4, s. P22-Artikkel i tidsskrift (Fagfellevurdert)

We prove that for arbitrary partitions lambda subset of kappa, and integers 0 <= c < r <= n, the sequence of Schur polynomials S(kappa+k.1c)/(lambda+k.1r)(x(1), ... , x(n)) for k sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.

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• 43.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stretched skew Schur polynomials are recurrent2014Inngår i: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 122, s. 1-8Artikkel i tidsskrift (Fagfellevurdert)

We show that sequences of skew Schur polynomials obtained from stretched semi-standard Young tableauxsatisfy a linear recurrence, which we give explicitly.Using this, we apply this to finding certain asymptotic behavior of these Schur polynomials and present conjectures on minimal recurrences for stretched Schur polynomials.

• 44.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
On Eigenvalues of the Schrödinger Operator with a Complex-Valued Polynomial Potential2012Inngår i: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 12, nr 1, s. 119-144Artikkel i tidsskrift (Fagfellevurdert)

We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation.

We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.

The first results can be derived from H. Habsch, while the case of a disconnected parameter space is new.

• 45.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
The cyclic sieving phenomenon on circular Dyck paths2019Inngår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 26, nr 4, artikkel-id P4.16Artikkel i tidsskrift (Fagfellevurdert)

We give a q-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this q-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Mobius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.

• 46.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Around multivariate Schmidt-Spitzer theorem2014Inngår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 446, s. 356-368Artikkel i tidsskrift (Fagfellevurdert)

Given an arbitrary complex-valued infinite matrix $\infmatA=(a_{ij}),$$i=1,\dotsc,\infty;$ $j=1,\dotsc,\infty$  and a positive integer $n$ we introduce anaturally associated  polynomial basis $\polybasis_\infmatA$ of$\C[x_0,\dotsc,x_n]$.We discuss some properties of the locus of  common zeros of all polynomials in $\polybasis_A$ having  a given degree $m$; the latter locus can beinterpreted as the spectrum of the $m\times (m+n)$-submatrix of $\infmatA$ formed by its  $m$ first rows and$(m+n)$ first columns. We initiate the study of the asymptotics of these spectra when $m\to \infty$ inthe case when $\infmatA$ is a banded Toeplitz matrix.In particular, we present and partially prove a conjectural multivariate analogof the well-known Schmidt-Spitzer theorem which describes  the spectral asymptotics for the sequence of principal minors of an arbitrarybanded Toeplitz matrix.Finally, we discuss relations between polynomial bases $\polybasis_\infmatA$ andmultivariate  orthogonal polynomials.

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• 47.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Discriminants, Symmetrized Graph monomials and Sums of Squares2012Inngår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 21, nr 4, s. 353-361Artikkel i tidsskrift (Fagfellevurdert)

In 1878, motivated by the requirements of the invariant the-ory of binary forms, J. J. Sylvester constructed, for every graphwith possible multiple edges but without loops, its symmetrizedgraph monomial, which is a polynomial in the vertex labels ofthe original graph. We pose the question for which graphs thispolynomial is nonnegative or a sum of squares. This problem ismotivated by a recent conjecture of F. Sottile and E. Mukhin onthe discriminant of the derivative of a univariate polynomial andby an interesting example of P. and A. Lax of a graph with fouredges whose symmetrized graph monomial is nonnegative butnot a sum of squares. We present detailed information about sym-metrized graph monomials for graphs with four and six edges,obtained by computer calculations.

• 48.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Modeling Realized Covariance of Asset Returns2019Licentiatavhandling, med artikler (Annet vitenskapelig)

In this thesis, which consists of two papers, we consider the modeling of positive definitive symmetric matrices, in particular covariance matrices of financial asset returns. The return covariance matrix describes the magnitude in which prices of financial assets tend to change over time, and how price changes between different assets are related. It is an instrumental quantity in many financial applications, and furthermore, an important component in understanding the dynamics present prior to and during times of financial turbulence, such as the 2008 financial crisis.

In the first paper, we provide several goodness-of-fit tests applicable to models driven by a centralized Wishart process. To apply such a distributional assumption has become a popular way of modeling the stochastic properties of time-series of realized covariance matrices for asset returns. The paper includes a simulation study that aims to investigate how the tests perform under model uncertainty stemming from parameter estimation. In addition, the presented methods are used to evaluate the fit of a typical model of realized covariance adapted to real data on six stocks traded on the New York Stock Exchange.

The second paper considers positive definite and symmetric random matrices of the exponential family. Under certain conditions for this class of distributions, we derive the Stein-Haff identity. Furthermore, we determine this identity in the case of the matrix-variate gamma distribution and apply it in order to present an estimator that outperforms the maximum likelihood estimator in terms of Stein's loss function. Finally, a small simulation study is conducted to support the theoretical results.

• 49.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stein-Haff Identity for the Exponential Family2018Inngår i: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 99, s. 7-18Artikkel i tidsskrift (Fagfellevurdert)

In this paper, the Stein-Haff identity is established for positive-definite and symmetric random matrices belonging to the exponential family. The identity is then applied to the matrix-variate gamma distribution, and an estimator that dominates the maximum likelihood estimator in terms of Stein's loss is obtained. Finally, a simulation study is conducted in order to support the theoretical results.

• 50.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen. Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
Goodness-of-fit tests for centralized Wishart processes2019Inngår i: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415XArtikkel i tidsskrift (Fagfellevurdert)

In this paper we present several goodness-of-fit tests for the centralized Wishart process, a popular matrix-variate time series model used to capture the stochastic properties of realized covariance matrices. The new test procedures are based on the extended Bartlett decomposition derived from the properties of the Wishart distribution and allows to obtain sets of independently and standard normally distributed random variables under the null hypothesis. Several tests for normality and independence are then applied to these variables in order to support or to reject the underlying assumption of a centralized Wishart process. In order to investigate the influence of estimated parameters on the suggested testing procedures in the finite-sample case, a simulation study is conducted. Finally, the new test methods are applied to real data consisting of realized covariance matrices computed for the returns on six assets traded on the New York Stock Exchange.

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