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  • 1. Aas, E.
    et al.
    Ayyer, A.
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Potka, Samu
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps2019Ingår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 52, nr 35, artikel-id 355001Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a finite one-dimensional totally asymmetric simple exclusion process with four types of particles, {1, 0, 1, }, in contact with reservoirs. Particles of species 0 can neither enter nor exit the lattice, and those of species are constrained to lie at the first and last site. Particles of species 1 enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species 1 enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.

  • 2.
    Aas, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A Markov Process on Cyclic Words2014Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    The TASEP (totally asymmetric simple exclusion process) studied here is a Markov chain on cyclic words over the alphabet{1,2,...,n} given by at each time step sorting an adjacent pair of letters chosen uniformly at random. For example, from the word 3124 one may go to 1324, 3124, 3124, 4123 by sorting the pair 31, 12, 24, or 43.

    Two words have the sametype if they are permutations of each other. If we restrict TASEP to words of some particular type m we get an ergodic Markov chain whose stationary distribution we denote by ζm. Soζm (u) is the asymptotic proportion of time spent in the state u if the chain started in some word of type m. The distribution ζ is the main object of study in this thesis. This distribution turns out to have several remarkable properties, and alternative characterizations. It has previously been studied both from physical, combinatorial, and probabilitistic viewpoints.

    In the first chapter we give an extended summary of known results and results in this thesis concerning ζ. The new results are described (and proved) in detail in Papers I - IV.

    The new results in Papers I and II include an explicit formula for the value ofζat sorted words and a product formula for decomposable words. We also compute some correlation functions for ζ. In Paper III we study of a generalization of TASEP to Weyl groups. In Paper IV we study a certain scaling limit of ζ, finding several interesting patterns of which we prove some. We also study an inhomogenous version of TASEP, in which different particles get sorted at different rates, which generalizes the homogenous version in several aspects. In the first chapter we compute some correlation functions for ζ

  • 3.
    Aas, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Limit points of the iterative scaling procedure2014Ingår i: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338, Vol. 215, nr 1, s. 15-23Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The iterative scaling procedure (ISP) is an algorithm which computes a sequence of matrices, starting from some given matrix. The objective is to find a matrix 'proportional' to the given matrix, having given row and column sums. In many cases, for example if the initial matrix is strictly positive, the sequence is convergent. It is known that the sequence has at most two limit points. When these are distinct, convergence to these two points can be slow. We give an efficient algorithm which finds the limit points, invoking the ISP only on subproblems for which the procedure is convergent.

  • 4.
    Aas, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Stationary probability of the identity for the TASEP on a Ring2012Övrigt (Övrigt vetenskapligt)
    Abstract [en]

    Consider the following Markov chain on permutations of length n. At each time step we choose a random position. If the letter at that position is smaller than the letter immediately to the left (cyclically) then these letters swap positions. Otherwise nothing happens, corresponding to a loop in the Markov chain. This is the circular TASEP. We compute the average proportion of time the chain spends at the identity permutation (and, in greater generality, at sorted words). This answers a conjecture by Thomas Lam.

  • 5.
    Aas, Erik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    TASEP in any Weyl groupManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We investigate a Markov chain dened by Thomas Lam [6], whichgeneralizes the multi-type TASEP on a ring to any Weyl group. For groups of typeC we dene an analogue of the multiline queues of Ferrari and Martin (which com-pute the stationary distribution for the classical TASEP). While our constructiondoes not suce for nding the stationary distribution, the construction gives thestationary distribution of a certain projection of Lam's chain. Also, our approach isincremental, in the sense that the construction appears to t into a pattern of 'con-jugation matrices', which remains to be fully worked out. We conjecture an explicitformula for the partition function of the model. Finally, we prove a theorem for theclassical TASEP which ts into the picture of viewing TASEP in a permutation-freeway.

  • 6.
    Aas, Erik
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Continuous multiline queues and TASEPManuskript (preprint) (Övrigt vetenskapligt)
  • 7.
    Aas, Erik
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sjöstrand, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A product formula for the TASEP on a ring2016Ingår i: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 48, nr 2, s. 247-259Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For a random permutation sampled from the stationary distributionof the TASEP on a ring, we show that, conditioned on the event that the rstentries are strictly larger than the last entries, the order of the rst entries isindependent of the order of the last entries. The proof uses multi-line queues asdened by Ferrari and Martin, and the theorem has an enumerative combinatorialinterpretation in that setting.As an application we prove a conjecture of Lam and Williams concerningSchubert factors of the stationary probability of certain states.Finally, we present a conjecture for the case where the small and large entriesare not separated.

  • 8. Abuzyarova, Natalia
    et al.
    Hedenmalm, Håkan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Branch point area methods in conformal mapping2006Ingår i: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 99, s. 177-198Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The classical estimate of Bieberbach that vertical bar a(2)vertical bar <= 2 for a given univalent function phi(z) = z + a(2)z(2) +... in the class S leads to the best possible pointwise estimates of the ratio phi''(z)/phi'(z) for phi is an element of S, first obtained by K oe be and Bieberbach. For the corresponding class E of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of psi(z)/psi'(z) for psi is an element of Sigma. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K oe be-Bieberbach estimate, is firmly rooted in area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.

  • 9. Acker, Andrew
    et al.
    Poghosyan, Michael
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Convex configurations for solutions to semilinear elliptic problems in convex rings2006Ingår i: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 31, nr 9, s. 1273-1287Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For a given convex ring Omega = Omega(2)\(Omega) over bar (1) and an L-1 function f : Omega x R -> R+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to Delta(p)u = f(x, u) in Omega u = 0 on partial derivative Omega(2) u = M on partial derivative Omega(1) with {x is an element of Omega : u(x) > s} boolean OR Omega(1) convex, for all s is an element of (0, M).

  • 10. Adiprasito, Karim
    et al.
    Björner, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Goodarzi, Afshin
    Freie Universität, Germany.
    Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals2017Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, nr 12, s. 3851-3865Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

  • 11. Adiprasito, Karim
    et al.
    Goodarzi, Afshin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Varbaro, Matteo
    Connectivity of pseudomanifold graphs from an algebraic point of view2015Ingår i: Comptes Rendus Mathematiques de l'Academie des Sciences = Mathematical reports of the academy of science, ISSN 0706-1994, Vol. 353, nr 12, s. 1061-1065Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Bjorner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.

  • 12.
    Adler, Jonas
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Data-driven Methods in Inverse Problems2019Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    In this thesis on data-driven methods in inverse problems we introduce several new methods to solve inverse problems using recent advancements in machine learning and specifically deep learning. The main goal has been to develop practically applicable methods, scalable to medical applications and with the ability to handle all the complexities associated with them.

    In total, the thesis contains six papers. Some of them are focused on more theoretical questions such as characterizing the optimal solutions of reconstruction schemes or extending current methods to new domains, while others have focused on practical applicability. A significant portion of the papers also aim to bringing knowledge from the machine learning community into the imaging community, with considerable effort spent on translating many of the concepts. The papers have been published in a range of venues: machine learning, medical imaging and inverse problems.

    The first two papers contribute to a class of methods now called learned iterative reconstruction where we introduce two ways of combining classical model driven reconstruction methods with deep neural networks. The next two papers look forward, aiming to address the question of "what do we want?" by proposing two very different but novel loss functions for training neural networks in inverse problems. The final papers dwelve into the statistical side, one gives a generalization of a class of deep generative models to Banach spaces while the next introduces two ways in which such methods can be used to perform Bayesian inversion at scale.

  • 13.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Lunz, Sebastian
    Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge, England..
    Banach Wasserstein GAN2018Ingår i: Advances in Neural Information Processing Systems 31 (NIPS 2018) / [ed] Bengio, S Wallach, H Larochelle, H Grauman, K CesaBianchi, N Garnett, R, Neural Information Processing Systems (NIPS) , 2018Konferensbidrag (Refereegranskat)
    Abstract [en]

    Wasserstein Generative Adversarial Networks (WGANs) can be used to generate realistic samples from complicated image distributions. The Wasserstein metric used in WGANs is based on a notion of distance between individual images, which induces a notion of distance between probability distributions of images. So far the community has considered l(2) as the underlying distance. We generalize the theory of WGAN with gradient penalty to Banach spaces, allowing practitioners to select the features to emphasize in the generator. We further discuss the effect of some particular choices of underlying norms, focusing on Sobolev norms. Finally, we demonstrate a boost in performance for an appropriate choice of norm on CIFAR-10 and CelebA.

  • 14.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta.
    Lunz, Sebastian
    Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
    Verdier, Olivier
    Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden ; Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway.
    Schönlieb, Carola-Bibiane
    Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Task adapted reconstruction for inverse problemsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

  • 15.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Ringh, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Karlsson, Johan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
    Learning to solve inverse problems using Wasserstein lossManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We propose using the Wasserstein loss for training in inverse problems. In particular, we consider a learned primal-dual reconstruction scheme for ill-posed inverse problems using the Wasserstein distance as loss function in the learning. This is motivated by miss-alignments in training data, which when using standard mean squared error loss could severely degrade reconstruction quality. We prove that training with the Wasserstein loss gives a reconstruction operator that correctly compensates for miss-alignments in certain cases, whereas training with the mean squared error gives a smeared reconstruction. Moreover, we demonstrate these effects by training a reconstruction algorithm using both mean squared error and optimal transport loss for a problem in computerized tomography.

  • 16.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Deep Bayesian InversionManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

  • 17.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Elekta Instrument AB, Stockholm, Sweden.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Learned Primal-Dual Reconstruction2018Ingår i: IEEE Transactions on Medical Imaging, ISSN 0278-0062, E-ISSN 1558-254X, Vol. 37, nr 6, s. 1322-1332Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose the Learned Primal-Dual algorithm for tomographic reconstruction. The algorithm accounts for a (possibly non-linear) forward operator in a deep neural network by unrolling a proximal primal-dual optimization method, but where the proximal operators have been replaced with convolutional neural networks. The algorithm is trained end-to-end, working directly from raw measured data and it does not depend on any initial reconstruction such as filtered back-projection (FBP). We compare performance of the proposed method on low dose computed tomography reconstruction against FBP, total variation (TV), and deep learning based post-processing of FBP. For the Shepp-Logan phantom we obtain >6 dB peak signal to noise ratio improvement against all compared methods. For human phantoms the corresponding improvement is 6.6 dB over TV and 2.2 dB over learned post-processing along with a substantial improvement in the structural similarity index. Finally, our algorithm involves only ten forward-back-projection computations, making the method feasible for time critical clinical applications.

  • 18.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Solving ill-posed inverse problems using iterative deep neural networks2017Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 33, nr 12, artikel-id 124007Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of 512 x 512 pixel images in about 0.4 s using a single graphics processing unit (GPU).

  • 19. Adler, M.
    et al.
    Chhita, S.
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    van Moerbeke, P.
    Tacnode GUE-minor processes and double Aztec diamonds2015Ingår i: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 162, nr 1-2, s. 275-325Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.

  • 20. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    van Moerbeke, Pierre
    Double Aztec diamonds and the tacnode process2014Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, s. 518-571Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Discrete and continuous non-intersecting random processes have given rise to critical "infinite-dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the tacnode process. It is also-shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.

  • 21. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    van Moerbeke, Pierre
    Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class2018Ingår i: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 21, nr 1, artikel-id 9Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.

  • 22. Aghajani, A.
    et al.
    Razani, Abdolrahman
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Department of Mathematics, Faculty of Science, Imam Khomeini International University, Iran .
    Some completeness theorems in the Menger probabilistic metric space2008Ingår i: Applied Sciences: APPS, ISSN 1454-5101, E-ISSN 1454-5101, Vol. 10, s. 1-8Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article, some new completeness theorems in probabilistic normed space are proved. Moreover, the existence of a constrictive Monger probabilistic normed space is shown.

  • 23. Agranovsky, M.
    et al.
    Khavinson, D.
    Shapiro, Harold
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Malmheden's theorem revisited2010Ingår i: Expositiones mathematicae, ISSN 0723-0869, E-ISSN 1878-0792, Vol. 28, nr 4, s. 337-350Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In 1934 Malmheden [16] discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin (1957) [8] 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in R-n.

  • 24.
    Ahlqvist, Eric
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Operations on Étale Sheaves of Sets2016Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Rydh visade 2011 att varje oramifierad morfi ƒ av algebraiska rum (algebraiska stackar) har en kanonisk och universell faktorisering genom ett algebraiskt rum (algebraisk stack) som han kallar den étala omslutningen av ƒ, där den första morfin är en sluten immersion och den andra är étale. Vi visar att då ƒ är étale så kan den étala omslutningen beskrivas genom att applicera vänsteradjunkten till tillbakadragningen av ƒ på den konstanta kärven som definieras av en punkterad mängd med två element. Då ƒ är en monomorfi, lokalt av ändlig typ så har vi en liknande beskrivning i termer av framtryckning med propert stöd.

  • 25. Aka, Menny
    et al.
    Breuillard, Emmanuel
    Rosenzweig, Lior
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    de Saxce, Nicolas
    On metric Diophantine approximation in matrices and Lie groups2015Ingår i: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 353, nr 3, s. 185-189Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the Diophantine exponent of analytic submanifolds of m x n real matrices, answering questions of Beresnevich, Kleinbock, and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over Q. We then apply these results to the determination of the Diophantine exponent of rational nilpotent Lie groups.

  • 26.
    Akbary, Amir
    et al.
    Univ Lethbridge, Dept Math & Comp Sci, 4401 Univ Dr, Lethbridge, AB T1K 3M4, Canada..
    Parks, James
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Univ Lethbridge, Dept Math & Comp Sci, 4401 Univ Dr, Lethbridge, AB T1K 3M4, Canada.
    On the Lang-Trotter conjecture for two elliptic curves2019Ingår i: Ramanujan Journal, ISSN 1382-4090, Vol. 49, nr 3, s. 585-623Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Following Lang and Trotter, we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over Q. In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the l-adic limits determining the l-factors of the universal constant, unlike the Lang-Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239-268, 2004), following the work of David et al. (Math Ann 368(1-2):685-752, 2017).

  • 27. Alberts, Tom
    et al.
    Binder, Ilia
    Viklund, Fredrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A Dimension Spectrum for SLE Boundary Collisions2016Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 343, nr 1, s. 273-298Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider chordal SLE curves for , where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension . We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

  • 28.
    Aleksanyan, Gohar
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Analysis of blow-ups for the double obstacle problem in dimension twoManuskript (preprint) (Övrigt vetenskapligt)
  • 29.
    Aleksanyan, Gohar
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Optimal regularity in the optimal switching problem2016Ingår i: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles{min⁡(−Δu1+f1,u1−u2+ψ1)=0min⁡(−Δu2+f2,u2−u1+ψ2)=0 arising in the optimal switching problem with two modes. We derive the optimal C1,1-regularity for the minimal solution under the assumption that the zero loop set L:={ψ1+ψ2=0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C1,1-regularity does not hold without the assumption L=L0‾. 

  • 30.
    Aleksanyan, Gohar
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Regularity of the free boundary in the biharmonic obstacle problemManuskript (preprint) (Övrigt vetenskapligt)
  • 31.
    Aleksanyan, Gohar
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Regularity results in free boundary problems2016Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis consists of three scientific papers, devoted to the regu-larity theory of free boundary problems. We use iteration arguments to derive the optimal regularity in the optimal switching problem, and to analyse the regularity of the free boundary in the biharmonic obstacle problem and in the double obstacle problem.In Paper A, we study the interior regularity of the solution to the optimal switching problem. We derive the optimal C1,1-regularity of the minimal solution under the assumption that the zero loop set is the closure of its interior.In Paper B, assuming that the solution to the biharmonic obstacle problem with a zero obstacle is suÿciently close-to the one-dimensional solution (xn)3+, we derive the C1,-regularity of the free boundary, under an additional assumption that the noncoincidence set is an NTA-domain.In Paper C we study the two-dimensional double obstacle problem with polynomial obstacles p1 p2, and observe that there is a new type of blow-ups that we call double-cone solutions. We investigate the existence of double-cone solutions depending on the coeÿcients of p1, p2, and show that if the solution to the double obstacle problem with obstacles p1 = −|x|2 and p2 = |x|2 is close to a double-cone solution, then the free boundary is a union of four C1,-graphs, pairwise crossing at the origin.

  • 32.
    Aleksanyan, Hayk
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Yerevan State University, Armenia.
    On greedy algorithm by renormed Franklin system2010Ingår i: East Journal on Approximations, ISSN 1310-6236, Vol. 16, nr 3, s. 273-296Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We characterize the all weighted greedy algorithms with respect to Franklin system which converge uniformly for continuous functions and almost everywhere for integrable functions. In case, when the algorithm fails to satisfy our classification criteria, we construct a continuous function for which the corresponding approximation diverges unboundedly almost everywhere. Some applications to wavelet systems are also discussed. 

  • 33.
    Aleksanyan, Hayk
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). The University of Edinburgh, UK.
    Regularity of boundary data in periodic homogenization of elliptic systems in layered media2017Ingår i: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 154, nr 1-2, s. 225-256Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by Gérard-Varet and Masmoudi (Acta Math. 209:133–178, 2012), and by Prange (SIAM J. Math. Anal. 45(1):345–387, 2012), however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connection.

  • 34.
    Aleksanyan, Hayk
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). University of Edinburgh, United Kingdom.
    Slow convergence in periodic homogenization problems for divergence-type elliptic operators2016Ingår i: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 48, nr 5, s. 3345-3382Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence-type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence-type elliptic equations set in halfspaces and with in finitely smooth data may converge to their corresponding boundary layer tails as slowly as one wishes depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence-type elliptic operators set in a bounded domain, and with all data being C-infinity-smooth, for which the boundary value homogenization holds with arbitrarily slow speed.

  • 35.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Karakhanyan, Aram
    The University of Edinburgh.
    K-surfaces with free boundaries2017Artikel i tidskrift (Refereegranskat)
  • 36.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Discrete balayage and boundary sandpile2016Ingår i: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538Artikel i tidskrift (Refereegranskat)
  • 37.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Shahgholian, Henrik
    Perturbed divisible sandpiles and quadrature surfaces2017Artikel i tidskrift (Refereegranskat)
  • 38.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Sjölin, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates2013Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 254, nr 6, s. 2626-2637Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely vertical bar u(epsilon)(x) - u(0)(x)vertical bar <= C-kappa epsilon((d-1)/2) 1/d(x)(kappa), for all x is an element of D, for all kappa > d - 1, where u(epsilon) and u(0) are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1 <= p < infinity we obtain L-P convergence rate as well.

  • 39.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sjölin, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Applications of Fourier Analysis in Homogenization of Dirichlet Problem III: Polygonal Domains2014Ingår i: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 20, nr 3, s. 524-546Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we prove convergence results for the homogenization of the Dirichlet problem for elliptic equations in divergence form with rapidly oscillating boundary data and non oscillating coefficients in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as convergence results. For larger exponents we prove that the convergence rate is close to optimal. We also suggest several directions of possible generalization of the results in this paper.

  • 40.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Sjölin, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L-p Estimates2015Ingår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 215, nr 1, s. 65-87Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

  • 41.
    Aleksanyan, Hayk
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). The University of Edinburgh.
    Shahgholian, Henrik
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sjölin, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    L2-estimates for singular oscillatory integral operators2016Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 441, nr 2, s. 529-548Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.

  • 42.
    Alexandersson, Per
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Polytopes and Large Counterexamples2019Ingår i: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 28, nr 1, s. 115-120Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand–Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer search. We also show that the multiset of hooks in a Young diagram is not enough information to determine the Ehrhart polynomial for an associated order polytope. This is somewhat counter-intuitive to the fact that the multiset of hooks always determine the leading coefficient of the Ehrhart polynomial.

  • 43.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Jordan, Linus
    Ecole Polytech Fed Lausanne, Dept Math, CH-1015 Lausanne, Switzerland..
    Enumeration of Border-Strip Decompositions and Weil-Petersson Volumes2019Ingår i: Journal of Integer Sequences, ISSN 1530-7638, E-ISSN 1530-7638, Vol. 22, nr 4, artikel-id 19.4.5Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We describe an injection from border-strip decompositions of certain diagrams to permutations. This allows us to provide enumeration results as well as q-analogues of enumeration formulas. Finally, we use this injection to prove a connection between the number of border-strip decompositions of the n x 2n rectangle and the Weil-Petersson volume of the moduli space of an n-punctured Riemann sphere.

  • 44.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Panova, Greta
    LLT polynomials, chromatic quasisymmetric functions and graphs with cycles2018Ingår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 341, nr 12, s. 3453-3482Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the noncircular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall-Littlewood polynomials. 

  • 45.
    Alexandersson, Per
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Sawhney, Mehtaab
    A Major-Index Preserving Map on Fillings2017Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 24, nr 4, artikel-id P4.3Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. Furthermore we define a similar variant of this map, that regards alternative models for the modified Macdonald polynomials at t = 0, and thus partially answers a question by J. Haglund. These maps together imply a certain uniqueness property regarding inversion- and coinversion-free fillings. These uniqueness properties allow us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux and the analogous question in the symmetric setting proves a conjecture by K. Nelson.

  • 46. Alm, Sven Erick
    et al.
    Janson, Svante
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Correlations for Paths in Random Orientations of G(n, p) and G(n, m)2011Ingår i: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 39, nr 4, s. 486-506Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study random graphs, both G(n, p) and G(n, m), with random orientations on the edges. For three fixed distinct vertices s, a, b we study the correlation, in the combined probability space, of the events {a -> s} and {s -> b}. For G(n, p), we prove that there is a p(c) = 1/2 such that for a fixed p < p(c) the correlation is negative for large enough n and for p > p(c) the correlation is positive for large enough n. We conjecture that for a fixed n >= 27 the correlation changes sign three times for three critical values of p. For G(n, m) it is similarly proved that, with p = m/((n)(2)), there is a critical p(c) that is the solution to a certain equation and approximately equal to 0.7993. A lemma, which computes the probability of non existence of any l directed edges in G(n, m), is thought to be of independent interest. We present exact recursions to compute P(a -> s) and P(a -> s, s -> b). We also briefly discuss the corresponding question in the quenched version of the problem.

  • 47. Alm, Sven Erick
    et al.
    Janson, Svante
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    First critical probability for a problem on random orientations in G(n,p)2014Ingår i: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 19, s. 69-Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the random graph G (n,p) with a random orientation. For three fixed vertices s, a, b in G(n,p) we study the correlation of the events {a -> s} (there exists a directed path from a to s) and {s -> b}. We prove that asymptotically the correlation is negative for small p, p < C-1/n, where C-1 approximate to 0.3617, positive for C-1/n < p < 2/n and up to p = p(2)(n). Computer aided computations suggest that p(2)(n) = C-2/n, with C-2 approximate to 7.5. We conjecture that the correlation then stays negative for p up to the previously known zero at 1/2; for larger p it is positive.

  • 48. Alm, Sven Erick
    et al.
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A Counter-Intuitive Correlation in a Random Tournament2011Ingår i: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 20, nr 1, s. 1-9Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Consider a randomly oriented graph G = (V, E) and let a, s and b be three distinct vertices in V. We study the correlation between the events {a -> s} and {s -> b}. We show that, counter-intuitively, when G is the complete graph K-n, n >= 5, then the correlation is positive. (It is negative for n = 3 and zero for n = 4.) We briefly discuss and pose problems for the same question on other graphs.

  • 49.
    Altafi, Nasrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The Weak Lefschetz Property of Equigenerated Monomial IdealsIngår i: Artikel i tidskrift (Övrigt vetenskapligt)
    Abstract [en]

    We determine the sharp lower bound for the Hilbert function in degree d of a

    monomial algebra failing the WLP over a polynomial ring with n variables and generated in

    degree d. We consider artinian ideals in the polynomial ring with

    n variables generated by homogeneous polynomials of degree d invariant under an action of

    the cyclic group Z/dZ. We give a complete classification of

    such ideals in terms of the WLP depending on the action.

  • 50. Aluffi, Paolo
    et al.
    Faber, Carel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Limits of PGL(3)-translates of plane curves, I2010Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 214, nr 5, s. 526-547Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the P-8 of 3 x 3 matrices to the P-N of plane curves of degree d. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in [P. Aluffi, C. Faber, Linear orbits of arbitrary plane curves, Michigan Math. J. 48(2000) 1-37].

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