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• 51. Aluffi, Paolo
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)

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].

• 52.
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.

• 53. 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.

• 54. 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.

• 55.
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.

• 56.
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.

• 57.
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.

• 58.
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.

• 59.
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.

• 60.
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.

• 61.
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.

• 62. 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).

• 63. 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.

• 64. 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.

• 65. 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.

• 66. 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.

• 67. 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.

• 68. 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.

• 69.
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.

• 70.
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.

• 71.
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.

• 72.
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.

• 73.
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.

• 74.
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.

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.

• 76.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Optimal regularity for the obstacle problem for the p-Laplacian2015Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, nr 6, s. 2167-2179Artikel i tidskrift (Refereegranskat)

In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p-2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p-1 in the spatial variables.

• 77.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Optimal regularity for the parabolic no-sign obstacle type problem2013Ingår i: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 15, nr 4, s. 477-499Artikel i tidskrift (Refereegranskat)

We study the parabolic free boundary problem of obstacle type Delta u - partial derivative u/partial derivative t = f chi({u not equal 0}). Under the condition that f = H nu for some function nu with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C-1 graph. Our result completes the theory for this type of problems for the heat operator.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The singular set of higher dimensional unstable obstacle type problems2013Ingår i: Rendiconti Lincei - Matematica e Applicazioni, ISSN 1120-6330, E-ISSN 1720-0768, Vol. 24, nr 1, s. 123-146Artikel i tidskrift (Refereegranskat)
• 79.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Global solutions of the obstacle problem in half-spaces, and their impact on local stability2005Ingår i: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 23, nr 3, s. 271-279Artikel i tidskrift (Refereegranskat)

We show that there are an abundance of non-homogeneous global solutions to the obstacle problem, in the half-space, Delta u = X-{u > 0}, u >= 0 inR(+)(2), with a (fixed) homogeneous boundary condition u(0, x(2)) = lambda(2) (x(2)(+))(2) (0 < lambda < 1/root 2) As a consequence we obtain local instability of the free boundary under C-1,C-1 perturbation, of the Dirichlet data.

• 80.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Equilibrium points of a singular cooperative system with free boundary2015Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, s. 743-771Artikel i tidskrift (Refereegranskat)

In this paper we initiate the study of maps minimising the energy integral(D)(vertical bar del u vertical bar(2) + 2 vertical bar u vertical bar) dx, which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Delta u = u / vertical bar u vertical bar chi({vertical bar u vertical bar >0}), u = (u(1,) ... ,u(m)) Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Double Obstacle Problems with Obstacles Given by Non-C-2 Hamilton-Jacobi Equations2012Ingår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 206, nr 3, s. 779-819Artikel i tidskrift (Refereegranskat)

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton-Jacobi equations that are not C (2). When the Hamilton-Jacobi equation is not C (2) then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C (1)-solutions to the Hamilton-Jacobi equation <Equation ID="Equa"> <MediaObject> </MediaObject> </Equation>, are, in fact, C (1,alpha/2), provided that . This result is optimal and, to the authors' best knowledge, new.

• 82.
Mathematics Institute, University of Warwick.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Mathematical Institute of the Heinrich Heine University.
On the singularities of a free boundary through Fourier expansion2012Ingår i: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 187, nr 3, s. 535-587Artikel i tidskrift (Refereegranskat)

In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Uniform Regularity Close to Cross Singularities in an Unstable Free Boundary Problem2010Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 296, nr 1, s. 251-270Artikel i tidskrift (Refereegranskat)

We introduce a new method for the analysis of singularities in the unstable problem Delta u = chi{u> 0}, which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neighborhood of the set at which the second derivatives of u are unbounded, the level set {u = 0} consists of two C-1-curves meeting at right angles. It is important that our result is not confined to the minimal solution of the equation but holds for all solutions.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
On the geometry and topology of initial data sets with horizons2018Ingår i: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 22, nr 5, s. 863-882Artikel i tidskrift (Refereegranskat)

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M, g,K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M \ ∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension n = 3, to higher dimensions.

• 85.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
A NEW TENSORIAL CONSERVATION LAW FOR MAXWELL FIELDS ON THE KERR BACKGROUND2017Ingår i: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 105, nr 2, s. 163-176Artikel i tidskrift (Refereegranskat)

A new, conserved, symmetric tensor field for a source-free Maxwell test field on a four-dimensional spacetime with a conformal Killing-Yano tensor, satisfying a certain compatibility condition, is introduced. In particular, this construction works for the Kerr spacetime.

• 86.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Constructing Multidimensional Dynamical Systems with Positive Lyapunov Exponents2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

I denna uppsats anpassar vi Marcelo Vianas generella konstruktion av  glatta transformationer som uppvisar icke-likformig expansion i flera  dimensioner. I vår konstruktion, istället för att koppla ihop vårt system med en kvadratisk avbildning, får vi expansionen från en avbildning med en  kubisk kritisk punkt. Dessutom så diskuterar vi hur argumentationen kan  utvidgas och expansionen fås av vissa typer av avbildningar med en kritisk punkt av godtyckligt hög (udda) grad. Vi diskuterar också problemen i att försöka använda Vianas metod med den så kallade "double standard"-avbildningen som källan till den icke-likformiga expansionen.

• 87.
Chalmers.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Proof of the cosmic no-hair conjecture in the T3-Gowdy symmetric Einstein-Vlasov setting2016Ingår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 7, s. 1565-1650Artikel i tidskrift (Refereegranskat)
• 88. Arakelyan, A.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Two-and multi-phase quadrature surfaces2017Ingår i: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 16, nr 6, s. 2023-2045Artikel i tidskrift (Refereegranskat)

In this paper we shall initiate the study of the two-and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation where dsx is the surface measure, μ = μ+-μ-is given measure with support in (a priori unknown domain) ω = ω+ [ω-, g is a given smooth positive function, and the integral holds for all functions h, which are harmonic on ω. Our approach is based on minimization of the corresponding two-and multiphase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.

• 89.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The Finite Difference Method for Two-Phase Parabolic Obstacle-Like Problem: Like ProblemArtikel i tidskrift (Övrigt vetenskapligt)

In this paper for two-phase parabolic obstacle-like problem, $\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,$ where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain, we will introduce a certain variational form, which allows us to define a notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of discretized scheme to a unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.

• 90.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The Finite Difference Methods for Multi-phase Free Boundary Problems2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consist of an introduction and four research papers concerning numerical analysis for a certain class of free boundary problems.

Paper I is devoted to the numerical analysis of the so-called two-phase membrane problem. Projected Gauss-Seidel method is constructed. We prove general convergence of the algorithm as well as obtain the error estimate for the finite difference scheme.

In Paper II we have improved known results on the error estimates for a Classical Obstacle (One-Phase) Problem with a finite difference scheme.

Paper III deals with the parabolic version of the two-phase obstacle-like problem. We introduce a certain variational form, which allows us to definea notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed.

In the last paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support. The proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions.

• 91.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference MethodManuskript (preprint) (Övrigt vetenskapligt)
• 92.
NAS Armenia, Inst Math, Yerevan 0019, Armenia..
Amer Univ Armenia, Inst Math, NAS Armenia, Yerevan 0019, Armenia.. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Qatar Univ, Dept Math Stat & Phys, POB 2713, Doha, Qatar..
Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles2019Ingår i: Bulletin of the Iranian Mathematical Society, ISSN 1018-6301, E-ISSN 1017-060X, Vol. 45, nr 1, s. 59-73Artikel i tidskrift (Refereegranskat)

In this paper, we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper, we present and discuss computational results.

• 93.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem2011Ingår i: Journal of Contemporary Mathematical Analysis, ISSN 1068-3623, Vol. 46, nr 3, s. 131-141Artikel i tidskrift (Refereegranskat)

In this paper we consider the finite difference scheme approximation for one-phase obstacle problem and obtain an error estimate for this approximation.

• 94.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Numerical Algorithms for a Variational problem of the Spatial Segregation of Reaction-diffusion SystemsArtikel i tidskrift (Övrigt vetenskapligt)
• 95. Are, Sasanka
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles2009Ingår i: Chinese Annals of Mathematics. Series B, ISSN 0252-9599, E-ISSN 1860-6261, Vol. 30, nr 6, s. 653-682Artikel i tidskrift (Refereegranskat)

Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting paxticles oil a surface, at a detailed atomistic level. However such algorithms are typically computationally expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-gained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.

• 96.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Fuzzy Riemann surfaces2009Ingår i: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, nr 6, s. 047-Artikel i tidskrift (Refereegranskat)

We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C (onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.

• 97. Arnlind, Joakim
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Goldfish geodesics and hamiltonian reduction of matrix dynamics2008Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 84, nr 1, s. 89-98Artikel i tidskrift (Refereegranskat)

We describe the Hamiltonian reduction of a time-dependent real-symmetric NxN matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars-Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.

• 98. Arnlind, Joakim
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Noncommutative Minimal Surfaces2016Ingår i: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 106, nr 8, s. 1109-1129Artikel i tidskrift (Refereegranskat)

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass representation.

• 99. Arnlind, Joakim
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Sogang University, South Korea .
The world as quantized minimal surfaces2013Ingår i: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 723, nr 4-5, s. 397-400Artikel i tidskrift (Refereegranskat)

It is pointed out that the equations Sigma(d)(i=1)[X-i, [X-i, X-j]] = 0 (and its super-symmetrizations, playing a central role in M-theory matrix models) describe non-commutative minimal surfaces - and can be solved as such.

• 100.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Max Planck Institute for Gravitational Physics.
Multi linear formulation of differential geometry and matrix regularizations2012Ingår i: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 91, nr 1, s. 1-39Artikel i tidskrift (Refereegranskat)

We prove that many aspects of the differential geometry of em-bedded Riemannian manifolds can be formulated in terms of multilinear algebraic structures on the space of smooth functions. Inparticular, we find algebraic expressions for Weingarten’s formula,the Ricci curvature and the Codazzi-Mainardi equations.For matrix analogues of embedded surfaces we define discretecurvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressionsfor the discrete Gauss curvature in terms of matrices representingthe embedding coordinates, and explicit examples are provided.Furthermore, we illustrate the fact that techniques from differen-tial geometry can carry over to matrix analogues by proving thata bound on the discrete Gauss curvature implies a bound on theeigenvalues of the discrete Laplace operator.

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