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• 151.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Technion Israel Inst Technol, Dept Math.
On the K-divisibility constant for some special finite-dimensional Banach couples2009Ingår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 360, nr 1, s. 130-155Artikel i tidskrift (Refereegranskat)

We prove new estimates of the K-divisibility constants for some special Banach couples. In particular, we prove that the K-divisibility constant for a couple of the form (U circle plus V, U) where U and V are non-trivial Hilbert spaces equals 2/root 3. We also prove estimates for the K-divisibility constant of the two-dimensional version of the couple (L-2, L-infinity), proving in particular that this couple is not exactly K-divisible. There are also several auxiliary results, including some estimates for relative Calderon constants for finite-dimensional couples.

• 152.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Berezin Transform in Polynomial Bergman Spaces2010Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 63, nr 12, s. 1533-1584Artikel i tidskrift (Refereegranskat)

Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.

• 153. Ameur, Yacin
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
FLUCTUATIONS OF EIGENVALUES OF RANDOM NORMAL MATRICES2011Ingår i: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 159, nr 1, s. 31-81Artikel i tidskrift (Refereegranskat)

In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane-the "droplet." We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.

• 154. Ameur, Yacin
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Random normal matrices and ward identities2015Ingår i: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 43, nr 3, s. 1157-1201Artikel i tidskrift (Refereegranskat)

We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman's solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

• 155.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Interpolation classes and matrix monotone functions2007Ingår i: Journal of operator theory, ISSN 0379-4024, E-ISSN 1841-7744, Vol. 57, nr 2, s. 409-427Artikel i tidskrift (Refereegranskat)

An interpolation function of order n is a positive function -/+ on (0, infinity) such that vertical bar vertical bar -/+ (A)(1/2) T -/+ (A)-(1/2) vertical bar vertical bar <= max(vertical bar vertical bar T vertical bar vertical bar, vertical bar A(1/2)TA(-1/2) vertical bar vertical bar) for all n x ii matrices T and A such that A is positive definite. By a theorem of Donoghue, the class C-n of interpolation functions of order n coincides with the class of functions -/+ such that for each n-subset S = {lambda i}(n)(i=1)of (0,infinity) there exists a positive Pick function h on (0, co) interpolating -/+ at S. This note comprises a study of the classes C-n and their relations to matrix monotone functions of finite order. We also consider interpolation functions on general unital C*-algebras.

• 156.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Combinatorics and zeros of multivariate polynomials2019Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate \$d\$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

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• 157.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Equidistributions of mahonian statistics over pattern avoiding permutations2018Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, nr 1, artikel-id P1.7Artikel i tidskrift (Refereegranskat)

A Mahonian d-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern functions of length at most d. Babson and Ste- ingrímsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over permutations in Sn avoiding a single classical pattern in S3. Tools used include block decomposition, Dyck paths and generating functions.

• 158.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2019Ingår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 50, nr 2, s. 165-190Artikel i tidskrift (Refereegranskat)

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Branden). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

• 159.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2018Ingår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Artikel i tidskrift (Refereegranskat)

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

• 160.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Stable multivariate generalizations of matching polynomialsManuskript (preprint) (Övrigt vetenskapligt)

The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan d-coverings, Hall, Puder and Sawin introduced the d-matching polynomial of a graph G, defined as the uniform average of matching polynomials over the set of d-sheeted covering graphs of G. We prove that a natural multivariate version of the d-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the d-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.

• 161.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The cone of cyclic sieving phenomena2019Ingår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, nr 6, s. 1581-1601Artikel i tidskrift (Refereegranskat)

We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

• 162.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Non-representable hyperbolic matroids2018Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 334, s. 417-449Artikel i tidskrift (Refereegranskat)

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vamos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids. We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vamos matroid and the generalized Vamos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre Turan inequality and an inequality due to Jensen. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

• 163. Ammann, B.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The conformal Yamabe constant of product manifolds2013Ingår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 141, nr 1, s. 295-307Artikel i tidskrift (Refereegranskat)

Let (V, g) and (W, h) be compact Riemannian manifolds of dimension at least 3. We derive a lower bound for the conformal Yamabe constant of the product manifold (V × W, g + h) in terms of the conformal Yamabe constants of (V, g) and (W, h).

• 164. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Mass endomorphism, surgery and perturbations2014Ingår i: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 64, nr 2, s. 467-487Artikel i tidskrift (Refereegranskat)

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

• 165. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Harmonic spinors and local deformations of the metric2011Ingår i: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 18, nr 5, s. 927-936Artikel i tidskrift (Refereegranskat)

Let (M, g) be a compact Riemannian spin manifold. The Atiyah-Singer index theorem yields a lower bound for the dimension of the kernel of the Dirac operator. We prove that this bound can be attained by changing the Riemannian metric g on an arbitrarily small open set.

• 166. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Low-dimensional surgery and the Yamabe invariant2015Ingår i: Journal of the Mathematical Society of Japan, ISSN 0025-5645, E-ISSN 1881-1167, Vol. 67, nr 1, s. 159-182Artikel i tidskrift (Refereegranskat)

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k <= n - 3. The smooth Yamabe invariants sigma(M) and sigma(N) satisfy sigma(N) >= min(sigma(M), Lambda) for a constant Lambda > 0 depending only on n and k. We derive explicit positive lower bounds for A in dimensions where previous methods failed, namely for (n, k) is an element of {(4, 1), (5, 1), (5, 2), (6, 3), (9, 1), (10, 1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

• 167. Ammann, Bernd
Smooth yamabe invariant and surgery2013Ingår i: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 94, nr 1, s. 1-58Artikel i tidskrift (Refereegranskat)

We prove a surgery formula for the smooth Yamabe invariant sigma(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Lambda(n), depending only on the dimension n of M, such that sigma(N) >= min{sigma(M), Lambda(n)}.

• 168. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Square-integrability of solutions of the Yamabe equation2013Ingår i: Communications in analysis and geometry, ISSN 1019-8385, E-ISSN 1944-9992, Vol. 21, nr 5, s. 891-916Artikel i tidskrift (Refereegranskat)

We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds, which are bounded and L-p for p = 2n/(n -2) are also L-2. This L-p-L-2 implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in our paper [4]. As an application we see that the smooth Yamabe invariant of any two-connected compact seven-dimensional manifold is at least 74.5. Similar conclusions follow in dimension 8 and in dimensions >= 11.

• 169. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Surgery and harmonic spinors2009Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 220, nr 2, s. 523-539Artikel i tidskrift (Refereegranskat)

Let M he a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.

• 170. Ammann, Bernd
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Surgery and the Spinorial tau-Invariant2009Ingår i: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 34, nr 10, s. 1147-1179Artikel i tidskrift (Refereegranskat)

We associate to a compact spin manifold M a real-valued invariant (M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's sigma-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then (N) epsilon min{(M), n}, where n is a positive constant depending only on n=dim M. Various topological conclusions can be drawn, in particular that is a spin-bordism invariant below n. Also, below n the values of cannot accumulate from above when varied over all manifolds of dimension n.

• 171.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
A thesis submitted in fulfilment of the requirements for the degree of Masters of Mathematics2015Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

An open problem introduced by J. Haglund was to find a bijective proof over Dyck paths that would interchange two of its statistics. This problem was known to be The Symmetry Problem of the q,t-Catalan polynomial and was proven by other means to be true. This project is an attempt to find a bijection, where we provide the bijection's behaviour under certain constrains. Then, we introduce an attempt to translate the problem from Dyck paths to other combinatorial structures. Finally we try to solve a related conjecture, called The Symmetry Problem of parking functions, which generalizes the previous problem. Some results we obtained from The Symmetry Problem of parking functions helped us characterize part of a bijective proof for Dyck paths.

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• 172.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
A comparison between different volatility models2011Självständigt arbete på avancerad nivå (magisterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
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• 173.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Approximate Bayesian Learning of Partition Directed Acyclic Graphs2016Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Partitionerade riktade acykliska grafer (engelska: PDAGs) är en modell där tabeller över betingade sannolikheter partitioneras i delar med lika sannolikhet. Detta gör att antalet parametrar som ska bestämmas kan reduceras, vilket i sin tur gör problemet beräkningsmässigt enklare. Ett känt samband mellan PDAGs och betecknade riktade acykliska grafer (engelska: LDAGs) sammanfattas här. Sedan jämförs en klustringsalgoritm med en algoritm som exakt bestämmer en PDAG. Klustringsalgoritmens pålitlighet kollas genom att använda den på simulerad data där det förväntade resultatet är känt.

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• 174.
KTH, Skolan för teknikvetenskap (SCI).
Morphisms of Fusion Systems2014Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

A fusion system on a finite group G with a Sylow p-subgroup P is a category on P with all subgroups of P as objects and group homomorphisms induced by conjugation in G as morphisms and was first introduced by L. Puig around 1990 in order to aid his research in finite group theory. The idea turned out to be fruitful and today, the theory of fusion systems is an active field in mathematics, with applications to topology, representation theory and finite group theory.

In this paper, we will, among other things, see how fusion systems aid in solving problems in finite group theory. We begin with an introduction to the theory with basic examples and then proceed to prove two famous theorems named after Burnside and Frobenius. However, to finish the proof of Frobenius’ theorem, we will require the focal subgroup theorem, whose proof requires transfer theory and is thus discussed. Afterwards, we introduce abstract and saturated fusion systems, in which one disregard the underlying group, and later prove that every fusion system on a finite group is saturated. We end with a discussion of morphisms of fusion systems, utilizing the concept previously developed, and generalize the isomorphism theorems to saturated fusion systems.

The presentation is well adapted for undergraduate students with limited knowledge of group and category theory and no previous knowledge of fusion systems is assumed.

Karl Amundsson kandidatexam
• 175.
KTH, Centra, Nordic Institute for Theoretical Physics NORDITA.
Dublin Inst Adv Studies, Sch Theoret Phys, 10 Burlington Rd, Dublin 4, Ireland.. Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England.;Univ Oxford, Math Inst, Radcliffe Observ Quarter, Andrew Wiles Bldg,Woodstock Rd, Oxford OX2 6GG, England.. Museo Stor Fis, Via Panisperna 89A, I-00184 Rome, Italy.;Ctr Studi & Ric Enrico Fermi, Via Panisperna 89A, I-00184 Rome, Italy.;Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via Marzolo 8, I-35131 Padua, Italy.;INFN, Sez Padova, Via Marzolo 8, I-35131 Padua, Italy.. Ctr Astron & Particle Theory, Univ Pk, Nottingham NG7 2RD, England.. Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England..
The mile high magic pyramid2019Ingår i: NONASSOCIATIVE MATHEMATICS AND ITS APPLICATIONS / [ed] Vojtechovsky, P Bremner, MR Carter, JS Evans, AB Huerta, J Kinyon, MK Moorhouse, GE Smith, JDH, AMER MATHEMATICAL SOC , 2019, s. 1-27Konferensbidrag (Refereegranskat)

Using a unified formulation of N = 1, 2, 4, 8, super Yang-Mills theories in D = 3 spacetime dimensions with fields valued respectively in R, C, H, O, it was shown that tensoring left and right multiplets yields a Freudenthal magic square of D = 3 supergravities. When tied in with the more familiar R, C, H, O description of super Yang-Mills in D = 3, 4, 6, 10 this results in a magic pyramid of supergravities: the known 4x4 magic square at the base in D = 3, a 3x3 square in D = 4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10.

• 176.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Improved Differential Diagnostics Using Methods in Machine Learning and Regression2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Det ﬁnns ett behov från både patienten och samhället att ha eﬀektiva verktyg för diﬀerentialdiagnostik. Matematiska förhållanden mellan sjukdomar och ob-serverbara konsekvenser deﬁnieras i uppsatsen. Speciﬁkt så används artiﬁciella neuronnät i modelleringen av läkarens metodik. För att föreslå ytterligare labprover eller symptom att leta efter inverteras nätverket genom att studera ett minimeringsproblem där målfunktionens gradient kan beräknas analytiskt. På grund av svårigheter i att erhålla verklig medicinsk data konstruerades ett program för att generera artiﬁciell patientdata. Denna patientdata kommer att användas för att etablera bevis på koncept. Några mått på kvalitén av patientdata deﬁnieras och används för att modellera nätverkets noggrannhet och träningstid. Det uppskattas sedan att ett problem med 4000 diagnoser och 20 000 observerbara konsekvenser skulle kräva 200 000 patienter för att uppnå en klassiﬁceringsnoggrannhet på 99%med en träningstid på 50 timmar beroende på beräkningskraften. I helhet verkar lösningsstrategin lovande men studier på verklig data krävs för deﬁnitiva svar.

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• 177.
KTH, Skolan för datavetenskap och kommunikation (CSC).
Modeling of a Cooling Airflow in an Electric Motor.2011Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

En elektrisk motor producerar mycket värme och kylningen av en sådan motor blir högst viktig. För att tillse ett luftflöde som uppfyller nedkylningskraven roterar en fläkt med motorn. Olyckligtvis uppfyller inte, typiskt sett, ett sådant system krav på buller. Detta arbete analyserar fläkten utifrån dess genererade buller och kapacitet och föreslår en förbättrad fläktdesign. Beräkningen av luftflödet utförs med mjukvaran COMSOL. Till en början sammanfattas översiktligt olika riktlinjer för fläktdesign och bullerkällor från fläkten. Nästföljande del beskriver det konkreta förloppet av simuleringen i COMSOL. Efter att dessa grundläggande frågor diskuterats utreds de olika bullerkällorna, specifikt bandbredd och tonljud, för fläkten och den förbättrade designen. Analys av olika geometriska parametrar utförs utifrån fläktkapaciteten, alltså det förflyttade luftflödet tillsammans med det genererade tryckskillnaderna. Till slut sammanfattas resultaten av dessa studier. Resultatet av detta blir att en förbättrad design påvisas.

• 178.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Smart Beta Investering Baserad på Makroekonomiska Indikatorer2015Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Den här uppsatsen undersöker möjligheten att hitta ett förhållande mellan Nasdaq Nordeas Smart Beta Index och en serie av makroekonomiska indikatorer. Detta förhållande kommer att implementeras som ett signalvärde i en portfölj bestående av alla sex index. För att se vilken påverkan signalvärdet har på en portföljs prestanda så undersöks tre portföljstrategier där den likaviktade portföljen används som riktmärke. Portföljerna ska omviktas varje månad och de portföljer som undersöks är mean-variance portföljen baserad på förhållandet med makroindikatorerna samt utan och den likaviktade portföljen baserad på förhållandet med makroindikatorerna.

För att kunna göra en prognos av portföljens prestanda anpassas en multivariat GARCH modell med tidsvarierande korrelationer till denna data. Tre stycken felfördelningar övervägs för modellen. Portföljens prestanda mäts både för prognosen samt för tiden med känd data och analysen är baserad på portföljernas Sharpe ratio.

Resultaten visar på att den portfölj som presterar bäst i tiden med känd data är meanvariance portföljen baserad på makroindikatorerna. I tiden för prognosen så är ingen port- följs Sharpe ratio statistiskt skild från den av den likaviktade portföljen.

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• 179.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Portfolio Performance Optimization Using Multivariate Time Series Volatilities Processed With Deep Layering LSTM Neurons and Markowitz2020Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Aktiemarknaden är en icke-linjär marknad, men många av de mest kända portföljoptimerings algoritmerna är baserad på linjära modeller. Under de senaste åren har den snabba utvecklingen inom maskininlärning skapat flexibla modeller som kan extrahera information ur komplexa mönster. I det här examensarbetet föreslår vi två sätt att optimera en portfölj, ett där ett neuralt nätverk utvecklas med avseende på multivariata tidsserier och ett annat där vi använder den linjära Markowitz modellen, där vi även lägger ett exponentiellt rörligt medelvärde på prisdatan. Ingångsdatan till vårt neurala nätverk är de dagliga slutpriserna, volymerna och marknadsindikatorer som t.ex. volatilitetsindexet VIX. Utgångsvariablerna kommer vara de predikterade priserna för nästa dag, som sedan bearbetas ytterligare för att producera mätvärden såsom förväntad avkastning, volatilitet och Sharpe ratio. LSTM-modellen producerar en portfölj med avkastning och risk som ligger närmre de verkliga marknadsförhållandena, men däremot gav resultatet ett högt felvärde och det visar att vår LSTM-modell är otillräckligt för att använda som ensamt predikteringssverktyg. Med det sagt så gav det ändå en bättre prediktion när det gäller trender än vad vi antog den skulle göra. Vår slutsats är därför att man bör använda flera neurala nätverk som indikatorer, där var och en är ansvarig för någon specifikt aspekt man vill analysera, och baserat på dessa dra en slutsats. Vårt resultat tyder också på att inmatningsdatan bör övervägas mera noggrant, eftersom predikteringsnoggrannheten.

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• 180.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
A mixed relaxed singular maximum principle for linear SDEs with random coefficientsArtikel i tidskrift (Refereegranskat)

We study singular stochastic control of a two dimensional stochastic differential equation, where the first component is linear with random and unbounded coefficients. We derive existence of an optimal relaxed control and necessary conditions for optimality in the form of a mixed relaxed-singular maximum principle in a global form. A motivating example is given in the form of an optimal investment and consumption problem with transaction costs, where we consider a portfolio with a continuum of bonds and where the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

• 181.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Contributions to the Stochastic Maximum Principle2009Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of four papers treating the maximum principle for stochastic control problems.

In the first paper we study the optimal control of a class of stochastic differential equations (SDEs) of mean-field type, where the coefficients are allowed to depend on the law of the process. Moreover, the cost functional of the control problem may also depend on the law of the process. Necessary and sufficient conditions for optimality are derived in the form of a maximum principle, which is also applied to solve the mean-variance portfolio problem.

In the second paper, we study the problem of controlling a linear SDE where the coefficients are random and not necessarily bounded. We consider relaxed control processes, i.e. the control is defined as a process taking values in the space of probability measures on the control set. The main motivation is a bond portfolio optimization problem. The relaxed control processes are then interpreted as the portfolio weights corresponding to different maturity times of the bonds. We establish existence of an optimal control and necessary conditons for optimality in the form of a maximum principle, extended to include the family of relaxed controls.

The third paper generalizes the second one by adding a singular control process to the SDE. That is, the control is singular with respect to the Lebesgue measure and its influence on the state is thus not continuous in time. In terms of the portfolio problem, this allows us to consider two investment possibilities - bonds (with a continuum of maturities) and stocks - and incur transaction costs between the two accounts.

In the fourth paper we consider a general singular control problem. The absolutely continuous part of the control is relaxed in the classical way, i.e. the generator of the corresponding martingale problem is integrated with respect to a probability measure, guaranteeing the existence of an optimal control. This is shown to correspond to an SDE driven by a continuous orthogonal martingale measure. A maximum principle which describes necessary conditions for optimal relaxed singular control is derived.

FULLTEXT01
• 182.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Necessary Optimality Conditions for Two Stochastic Control Problems2008Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of two papers concerning necessary conditions in stochastic control problems. In the first paper, we study the problem of controlling a linear stochastic differential equation (SDE) where the coefficients are random and not necessarily bounded. We consider relaxed control processes, i.e. the control is defined as a process taking values in the space of probability measures on the control set. The main motivation is a bond portfolio optimization problem. The relaxed control processes are then interpreted as the portfolio weights corresponding to different maturity times of the bonds. We establish existence of an optimal control and necessary conditions for optimality in the form of a maximum principle, extended to include the family of relaxed controls.

In the second paper we consider the so-called singular control problem where the control consists of two components, one absolutely continuous and one singular. The absolutely continuous part of the control is allowed to enter both the drift and diffusion coefficient. The absolutely continuous part is relaxed in the classical way, i.e. the generator of the corresponding martingale problem is integrated with respect to a probability measure, guaranteeing the existence of an optimal control. This is shown to correspond to an SDE driven by a continuous orthogonal martingale measure. A maximum principle which describes necessary conditions for optimal relaxed singular control is derived.

FULLTEXT01
• 183.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
The relaxed general maximum principle for singular optimal control of diffusions2009Ingår i: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, ISSN 01676911, Vol. 58, nr 1, s. 76-82Artikel i tidskrift (Refereegranskat)

In this paper we study optimality in stochastic control problems where the state process is a stochastic differential equation (SDE) and the control variable has two components, the first being absolutely continuous and the second singular. A control is defined as a solution to the corresponding martingale problem. To obtain existence of an optimal control Haussmann and Suo [U.G. Haussmann, W. Suo, Singular optimal stochastic controls I: Existence, SIAM J. Control Optim. 33 (3) (1995) 916-936] relaxed the martingale problem by extending the absolutely continuous control to the space of probability measures on the control set. Bahlali et al. [S. Bahlali, B. Djehiche, B. Mezerdi, The relaxed stochastic maximum principle in singular optimal control of diffusions, SIAM J. Control Optim. 46 (2) (2007) 427-444] established a maximum principle for relaxed singular control problems with uncontrolled diffusion coefficient. The main goal of this paper is to extend their results to the case where the control enters the diffusion coefficient. The proof is based on necessary conditions for near optimality of a sequence of ordinary controls which approximate the optimal relaxed control. The necessary conditions for near optimality are obtained by Ekeland's variational principle and the general maximum principle for (strict) singular control problems obtained in Bahlali and Mezerdi [S. Bahlali, B. Mezerdi, A general stochastic maximum principle for singular control problems, Electron J. Probab. 10 (2005) 988-1004. Paper no 30]. © 2008 Elsevier B.V. All rights reserved.

• 184.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
The relaxed stochastic maximum principle in singular optimal control of diffusions with controlled diffusion coefficientManuskript (Övrigt vetenskapligt)
• 185.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
A maximum principle for relaxed stochastic control of linear SDEs with application to bond portfolio optimization2010Ingår i: Mathematical Methods of Operations Research, ISSN 1432-2994, E-ISSN 1432-5217, Vol. 72, nr 2, s. 273-310Artikel i tidskrift (Refereegranskat)

We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity.

• 186.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
A maximum principle for SDEs of mean-field type2011Ingår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 63, nr 3, s. 341-356Artikel i tidskrift (Refereegranskat)

We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.

• 187.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Factors affecting the proportion of smartphone usage at Flygresor.se2017Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

fulltext
• 188.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Solving the Train Timetabling Problem by using Rapid Branching2016Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

The topic of this thesis is the implementation of rapid branching to find an integer solution for the train timetabling problem. The techniques that rapid branching are based on are presented. The important aspect of rapid branching are discussed and then the algorithm is applied to some artificial problems. It is shown that rapid branching can be both faster and slower than a standard integer solver depending on the problem instance. For the most realistic set of the examined instances, rapid branching turned out to be faster than the standard integer solver and produce satisficingly high quality solutions.

fulltext
• 189.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Missing Data in Value-at-Risk Analysis: Conditional Imputation in Optimal Portfolios Using Regression2013Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

A regression-based method is presented in order toregenerate missing data points in stock return time series. The method usesonly complete time series of assets in optimal portfolios, in which the returnsof the underlying tend to correlate inadequately with each other. The studyshows that the method is able to replicate empirical VaR-backtesting resultswhere all data are available, even when up to 90% of the time series in half ofthe assets in the portfolios have been removed.

fulltext
• 190.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Counting words avoiding patterns of length three with generating functions2015Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Mängden av 123-undvikande ord med exakt r förekomster av varje bokstav enumererades nyligen av N. Shar och D. Zeilberger. Detta arbete enumererar mer komplicerade mängder av mönsterundvikande ord, såsom de som tillåter 1,2, ... eller r förekomster av varje bokstav, eller de för vilka antalet förekomster av varje enskild bokstav följer en upprepande sekvens. Resultaten generaliseras även till att gälla alla mönster av längd tre med distinkta bokstäver. De genererande funktionerna som enumererar orden bevisas vara algebraiska, för alla undersökta ordmängder. Ett ansenligt antal koefficienter för de relevanta genererande funktionerna har beräknats och de första av dessa har bekräftats med oberoende metoder. Dessa koefficienters asymptotiska beteende har visats vara exponentiellt. Metoden som utnyttjas involverar en uppdelning i delord som tillåter konstruerandet av ekvationer som relaterar de genererande funktionerna.

fulltext
• 191.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
On Invertibility of the Radon Transform and Compressive Sensing2014Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis contains three articles. The first two concern inversion andlocal injectivity of the weighted Radon transform in the plane. The thirdpaper concerns two of the key results from compressive sensing.In Paper A we prove an identity involving three singular double integrals.This is then used to prove an inversion formula for the weighted Radon transform,allowing all weight functions that have been considered previously.Paper B is devoted to stability estimates of the standard and weightedlocal Radon transform. The estimates will hold for functions that satisfy an apriori bound. When weights are involved they must solve a certain differentialequation and fulfill some regularity assumptions.In Paper C we present some new constant bounds. Firstly we presenta version of the theorem of uniform recovery of random sampling matrices,where explicit constants have not been presented before. Secondly we improvethe condition when the so-called restricted isometry property implies the nullspace property.

Thesis
• 192.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Stockholm University.
Stability estimates with a priori bound for the inverse local Radon transformManuskript (preprint) (Övrigt vetenskapligt)

We consider the inverse problem for the 2-dimensional weighted local Radon transform $R_{m}[f]$, where $f$ is supported in $y\geq x^2$ and $R_{m}[f](\xi,\eta)=\int f(x,\xi x+\eta)m(x,\xi,\eta)dx$ is defined near $(\xi,\eta)=(0,0)$. For weight functions satisfying a certain differential equation we give weak estimates of$f$ in terms of $R_{m}[f]$ for functions $f$ that satisfies an a priori bound.

• 193.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
On the Theorem of Uniform Recovery of Random Sampling Matrices2014Ingår i: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 60, nr 3, s. 1700-1710Artikel i tidskrift (Refereegranskat)

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m greater than or similar to s(ln s)(3) ln N. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m x N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, delta s, ensuring sparse recovery via l(1)-minimization. We show that delta(2s) < 4/root 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type delta(2s) < 2/3.

• 194.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Locating Multiple Change-Points Using a Combination of Methods2014Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Syftet med studien är att hitta en metod som identifierar tidpunkterna för strukturella brott i en tidsserie med okända egenskaper. De metoder som undersöks är CUSUM och CUSUM av kvadrater, CUSUM test med OLS-residualer, Mann-Whitney-test samt Quandts log likelihood ratio. Eftersom alla metoder identifierar enbart en brytpunkt används binära uppdelningstekniken för att hitta multipla brytpunkter. Studien visar att CUSUM-test med OLS-residualer, Mann-Whitney-test och Quandt’s log likelihood ratio fungerar bra för de flesta stickproven medan CUSUM och CUSUM av kvadrater inte hittar tidpunkten för brytpunkterna. Vidare så visar studien att binära uppdelningstekniken fungerar bra med alla metoder och kan identifiera multipla brytpunkter i de flesta fallen. Studien visar också att resultaten för det mesta kan förbättras genom att använda en kombination av metoderna.

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• 195.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Free boundary problems and global solutions in half spaces2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
• 196.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
On the De Giorgi conjecture in half spacesManuskript (preprint) (Övrigt vetenskapligt)
• 197.
Max Planck Institute, Leipzig.
On the regularity of a free boundary near contact points with a fixed boundary2007Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 232, nr 1, s. 285-302Artikel i tidskrift (Refereegranskat)

We investigate the regularity of a free boundary near contact points with a fixed boundary, with C boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C 1 up to the fixed boundary. We will also construct some examples of irregular free boundaries.

• 198.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Optimal regularity for the Signorini problem and its free boundary2015Ingår i: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297Artikel i tidskrift (Refereegranskat)

We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if (Formula presented.) minimizes (Formula presented.)in the convex set (Formula presented.)where (Formula presented.) say. Then (Formula presented.). Moreover the free boundary, given by (Formula presented.)will be a (Formula presented.) graph close to points where (Formula presented.) is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

• 199.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
The free boundary near the fixed boundary for the heat equationManuskript (preprint) (Övrigt vetenskapligt)
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Optimal Regularity for the No-Sign Obstacle Problem2013Ingår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, nr 2, s. 245-262Artikel i tidskrift (Refereegranskat)

In this paper we prove the optimal C-1,C-1(B-1/2)-regularity for a general obstacle-type problem Delta u = f chi({u not equal 0}) in B-1, under the assumption that f * N is C-1,C-1(B-1), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C-1,C-1-regularity. As a by-product of the C-1,C-1-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x(0), the free boundary is locally a C-1-graph close to x(0) provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.

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