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Fri, 10 Nov 2017 09:47:24 +0100Fri, 10 Nov 2017 09:47:24 +0100Polynomial functions of modular lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060
A polynomial function \(f : L \to L\) of a lattice \(\mathcal{L}\) = \((L; \land, \lor)\) is generated by the identity function id \(id(x)=x\) and the constant functions \(c_a (x) = a\) (for every \(x \in L\)), \(a \in L\) by applying the operations \(\land, \lor\) finitely often. Every polynomial function in one or also in several variables is a monotone function of \(\mathcal{L}\).
If every monotone function of \(\mathcal{L}\)is a polynomial function then \(\mathcal{L}\) is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices
and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice \(\mathcal{L}\) with a finite interval is order-polynomially complete if and only if \(\mathcal{L}\) is finite projective geometry. If \(\mathcal{L}\) is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of \(\mathcal{L}\). In the last sections we show the descriptive power of polynomial functions of
lattices and present several applications in geometry.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060Fri, 10 Nov 2017 09:47:24 +0100On derived varieties
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059
Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In this paper this tool is developed further in order to use it for algebraic constructions of derived algebras. Especially the operator \(S\) of subalgebras, \(H\) of homomorphic irnages and \(P\) of direct products are studied. Derived groupoids from the groupoid \(N or (x,y)\) = \(x'\wedge y'\) and from abelian groups are considered. The latter class serves as an example for fluid algebras and varieties. A fluid variety \(V\) has no derived variety as a subvariety and is introduced as a counterpart for solid varieties. Finally we use a property of the commutator of derived algebras in order to show that solvability and nilpotency are preserved under derivation.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059Fri, 10 Nov 2017 09:22:23 +0100Error estimates for Tikhonov regularization with unbounded regularizing operators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056
It is shown that Tikhonov regularization for ill- posed operator equation
\(Kx = y\) using a possibly unbounded regularizing operator \(L\) yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach.M. Thamban Nairreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056Thu, 09 Nov 2017 12:01:16 +0100On the expected number of shadow vertices of the convex hull of random points
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5051
Let \(a_1,\dots,a_m\) be independent random points in \(\mathbb{R}^n\) that are independent and identically distributed spherically symmetrical in \(\mathbb{R}^n\). Moreover, let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_m\) and let \(L_k\) be an arbitrary \(k\)-dimensional
subspace of \(\mathbb{R}^n\) with \(2\le k\le n-1\). Let \(X_k\) be the orthogonal projection image of \(X\) in \(L_k\). We call those vertices of \(X\), whose projection images in \(L_k\) are vertices of \(X_k\)as well shadow vertices of \(X\) with respect to the subspace \(L_k\) . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of \(X\) in \(L_k\).Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5051Thu, 09 Nov 2017 10:49:33 +0100A comparison method for expectations of a class of continuous polytope functionals
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5047
Let \(a_1,\dots,a_n\) be independent random points in \(\mathbb{R}^d\) spherically symmetrically but not necessarily identically distributed. Let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_n\) and for any \(k\)-dimensional subspace \(L\subseteq \mathbb{R}^d\) let \(Vol_L(X) :=\lambda_k(L\cap X)\) be the volume of \(X\cap L\) with respect to the \(k\)-dimensional Lebesgue measure \(\lambda_k, k=1,\dots,d\). Furthermore, let \(F^{(i)}\)(t):= \(\bf{Pr}\) \(\)(\(\Vert a_i \|_2\leq t\)),
\(t \in \mathbb{R}^+_0\) , be the radial distribution function of \(a_i\). We prove that the expectation
functional \(\Phi_L\)(\(F^{(1)}, F^{(2)},\dots, F^{(n)})\) := \(E(Vol_L(X)\)) is strictly decreasing in
each argument, i.e. if \(F^{(i)}(t) \le G^{(i)}(t)t\), \(t \in {R}^+_0\), but \(F^{(i)} \not\equiv G^{(i)}\), we show \(\Phi\) \((\dots, F^{(i)}, \dots\)) > \(\Phi(\dots,G^{(i)},\dots\)). The proof is clone in the more general framework
of continuous and \(f\)- additive polytope functionals.Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5047Wed, 08 Nov 2017 13:33:06 +0100New Integrals for \(\zeta(s)\zeta(s+1)\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5038
Andreas Guthmannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5038Tue, 07 Nov 2017 11:46:13 +0100Derived Varieties of Semigroups and Groupoids
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4889
Dietmar Schweigert; S.L. Wismathreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4889Thu, 19 Oct 2017 11:35:28 +0200Determinantal Rational Surface Singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4866
In this paper we give explicit equations for determinantal rational surface singularities and prove dimension formulas for the \(T^1\) and \(T^2\) for those singularities.Theo de Jongreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4866Tue, 17 Oct 2017 11:38:01 +0200Superlinear convergence rates for the Lanczos method applied to elliptic operators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4864
This paper investigates the convergence of the Lanczos method for computing the smallest eigenpair of a selfadjoint elliptic differential operator via inverse iteration (without shifts).
Superlinear convergence rates are established, and their sharpness is investigated for a simple model problem. These results are illustrated numerically for a more difficult problem.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4864Tue, 17 Oct 2017 09:10:44 +0200Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4863
This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration.
These assumptions are fulfilled , e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4863Tue, 17 Oct 2017 08:51:54 +0200Nonstationary lterated Tikhonov Regularization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4862
A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. lt is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.Martin Hanke; C.W. Groetschreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4862Mon, 16 Oct 2017 16:20:02 +0200A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4861
The first part of this paper studies a Levenberg-Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fréchet derivative of the nonlinear operator at the exact solution, the new Levenberg-Marquardt scheme is suitable for ill-posed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and dornain
topologies. Estimates of this type are established in the second part of the paper for ill-posed parameter identification problems arising in inverse groundwater hydrology. Both, transient and steady state data are investigated. Finally, the numerical performance of the new Levenberg-Marquardt scheme is
studied and compared to a usual implementation on a realistic but synthetic 2D model problem from the engineering literature.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4861Mon, 16 Oct 2017 15:27:12 +0200Infinitesimal module deformations in the Thom-Sebastiani Problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4851
Florian Enescu; Gerhard Pfister; Dorin Popescureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4851Mon, 16 Oct 2017 09:24:50 +0200On the Solution Region for Certain Scheduling Problems with Preemption
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4839
The paper deals with parallel-machine and open-shop scheduling problems with preemptions and arbitrary nondecreasing objective function. An approach to describe
the solution region for these problems and to reduce them to minimization problems on polytopes is proposed. Properties of the solution regions for certain problems are investigated. lt is proved that open-shop problems with unit processing times are equivalent to certain parallel-machine problems, where preemption is allowed at arbitrary time. A polynomial algorithm is presented transforming a schedule of one type into a schedule of the other type.Heidemarie Bräsel; Natalia Shakhlevichreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4839Fri, 13 Oct 2017 14:01:28 +0200Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/802
Christian Brücker; Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/802Mon, 03 Apr 2000 00:00:00 +0200On moduli spaces of semiquasihomogeneous singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/805
Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/805Mon, 03 Apr 2000 00:00:00 +0200Semicontinuity for representations of Cohen-Macaulay rings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/807
Yurij Drozd; Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/807Mon, 03 Apr 2000 00:00:00 +0200Equianalytic and equisingular families of curves on surfaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808
Gert-Martin Greuel; Christoph Lossenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808Mon, 03 Apr 2000 00:00:00 +0200Moduli Spaces of Semiquasihomogeneous Singularities with fixed Principal Part
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/810
Gert-Martin Greuel; Claus Hertling; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/810Mon, 03 Apr 2000 00:00:00 +0200Geometric quotients of unipotent group actions II
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/813
Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/813Mon, 03 Apr 2000 00:00:00 +0200Tangent measure distributions of hyperbolic Cantor sets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/819
Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.Peter Mörters; Daniela Kriegpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/819Mon, 03 Apr 2000 00:00:00 +0200Moduli spaces of decomposable morhpisms of sheaves and quotients by non-reductive groups
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/827
We extend the methods of geometric invariant theory to actions of non reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non recutive. Given a linearization of the natural actionof the group Aut(E)xAut(F) on Hom(E,F), a homomorphism iscalled stable if its orbit with respect to the unipotentradical is contained in the stable locus with respect to thenatural reductive subgroup of the automorphism group. Weencounter effective numerical conditions for a linearizationsuch that the corresponding open set of semi-stable homomorphismsadmits a good and projective quotient in the sense of geometricinvariant theory, and that this quotient is in additiona geometric quotient on the set of stable homomorphisms.Jean Marc Drezed; Günther Trautmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/827Mon, 03 Apr 2000 00:00:00 +0200