Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.     

A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements

We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular, this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.     

The homotopy type of the complement of a coordinate subspace arrangement

The homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.     

Hilbert series of subspace arrangements

The vanishing ideal I of a subspace arrangement V₁∪V₂∪?∪V_m⊆V is an intersection I₁∩I₂∩?∩I_m of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J=I₁I₂?I_m without any assumptions about the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. We will also apply our results to generalized principal component analysis (GPCA), a tool that is useful for computer vision and image processing.     

Calculus in positive characteristic p

We revisit hyperderivatives to build on the integral theory of calculus in positive characteristic p. In particular, we give necessary and sufficient conditions for the exactness of a hyperdifferential form associated with hyperderivatives and then propose a closed formula for finding the hyperantiderivative of an exact form.     

General lexicographic shellability and orbit arrangements

We introduce a new poset property which we call EC-shellability. It is more general than the more established concept of EL-shellability, but it still implies shellability. Because of Theorem 3.10, EC-shellability is entitled to be called general lexicographic shellability. As an application of our new concept, we prove that intersection lattices Π_λ of orbit arrangementsA _λ are EC-shellable for a very large class of partitions λ. This allows us to compute the topology of the link and the complement for these arrangements. In particular, for this class of λs, we are able to settle a conjecture of Björner [B94, Conjecture 13.3.2], stating that the cohomology groups of the complement of the orbit arrangements are torsion-free. We also present a class of partitions for which Π_λ is not shellable, along with other issues scattered throughout the paper.     

On questions related to saturated chains of primes in polynomial rings

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X₁,…,X_n〉 residually Jaffard (resp., totally Jaffard) when R(X₁,…,X_n) is ? We construct a three-dimensional local ring R such that R(X₁,…,X_n) is totally Jaffard (and hence, residually Jaffard) whereas R〈X₁,…,X_n〉 is not residually Jaffard (and hence, not totally Jaffard).     

Zur Konvergenz des Rayleigh-Ritz-Verfahrens bei Eigenwertaufgaben

Zusammenfassung Bei der näherungsweisen Berechnung von Eigenwerten und Eigenelementen spielt das klassische Rayleigh-Ritz-Verfahren gerade im Hinblick auf die Finite-Elemente-Methode eine bedeutende Rolle. Umfangreiche Untersuchungen zur Konvergenz des Verfahrens liegen vor (vgl. etwa [1, 2, 4, 5, 7, 12, 15, 16, 19, 20]). Dabei werden zumeist quantitative Fehlerabschätzungen für konkrete Approximationen spezieller Probleme hergeleitet. Die Behandlung der Näherungseigenelemente erweist sich als besonders schwierig, wenn die zugehörigen Eigenwerte mehrfach vorliegen. Die für viele weiteren Untersuchungen fundamentalen Ergebnisse von Birkhoff et al. [1] gelten ausschließlich für einfache Eigenwerte.Das Hauptanliegen dieser Arbeit ist die Darlegung des qualitativen und quantitativen Konvergenzverhaltens der Näherungseigenelemente. In einer rein funktionalanalytischen Vorgehensweise betrachten wir Eigenwertaufgaben, die sich durch halbbeschränkte selbstadjungierte Operatoren beschreiben lassen. Die Konvergenzaussagen werden zurückgeführt auf die Approximationsgüte der Diskretisierung. Die erzielten Abschätzungen stehen somit generell fürjede Konkretisierung zur Verfügung. Insbesondere werden die Resultate aus [1] verallgemeinert. Die Beweise orientieren sich direkt an der Problemstellung. Methoden der Spektralapproximation (vgl. Chatelin [3], Vainikko [20]) werden nicht eingesetzt.

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On the convergence of the Rayleigh-Ritz method for eigenvalue problems

Summary This paper is concerned with the Rayleigh-Ritz method applied to eigenvalue problems, discribed by operators which are selfadjoint and bounded from below. In a purely functional analytic procedure the convergence results are reduced to error estimates for the discretization.

Dedicated to the memory of Professor Lothar Collatz     

Asymptotische Fehlerschranken für Rayleigh-Ritz-Approximationen selbstadjungierter Eigenwertaufgaben

Zusammenfassung In der vorliegenden Arbeit leiten wir Fehlerschranken her für die Rayleigh-Ritz-Näherungen der Eigenwerte und Eigenelemente von selbstadjungierten Eigenwertaufgaben, deren Spektren nach unten beschränkt und anfangsdiskret sind. Solche Abschätzungen haben gerade im Hinblick auf ihre Bedeutung bei der Finite-Elemente-Methode eine lange Tradition, vgl. etwa [3-5, 7-9] sowie insbesondere die in [3] aufgeführte Literatur.Verfeinerte Fehlerschranken haben kürzlich Babuka und Osborn [1–3] angegeben: Speziell für mehrfach vorliegende Eigenwerte führen sie Approximationsgrößen ein, die die Güte der Diskretisierung beschreiben, und schätzen dann in Termen dieser Größen die Fehler ab. Dabei betrachten sie jedoch aus beweistechnischer Notwendigkeit heraus stets den Fall eines rein diskreten Spektrums, das sich durch selbstadjungierte und kompakte Operatoren beschreiben läßt. Wir lösen uns hier von dieser Einschränkung und geben in analoger Weise das asymptotische Verhalten der Fehler auch für solche Aufgaben an, die ein wesentliches Spektrum besitzen. Zugleich verbessern wir dabei die in [7] aufgestellten Schranken. Unsere Beweismethoden liefern explizit alle in den Fehlerschranken auftretenden Konstanten. Diese ergeben sich allein aus dem Spektrum der betrachteten Aufgabe. Desweiteren können wir alle Beweise mit reellwertigem Skalarkörper durchführen, da wir eine Darstellung von Spektralprojektoren durch Kurvenintegrale (vgl. etwa die Methoden der Spektralapproximation bei Chatelin [5]) für selbstadjungierte Probleme nicht benötigen. Die erhaltenen Ergebnisse gelten jedoch auch entsprechend für komplexwertige Skalarkörper.

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Asymptotic error estimates for Rayleigh-Ritz-approximations of selfadjoint eigenvalue problems

Summary In this paper we establish estimates for the approximation of the eigenvalues and eigenvectors of a selfadjoint eigenvalue problem bounded below with spectrum that begins with isolated eigenvalues of finite multiplicity. Results on the asymptotic behavior of the errors recently proved by Babuka and Osborn [1–3] are also valid for problems with nontrivial essential spectrum.

Herrn Professor Dr. Harro Heuser zum 65. Geburtstag gewidmet     

Weitzenböck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation δ   of the polynomial algebra K[_Xd]=K[_x1,…,_xd]$K[X_d]=K[x_1\text{,}\dots \text{,}{x}_d]$ in several variables over a field K   of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[_Xd]^δ$K{[X_d]}^{\delta }$ (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation δ of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K  . Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ of the free metabelian Lie algebra _Ld/_Ld″$L_d/L_d″$ with d   generators. We show that the vector space of the constants (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ in the commutator ideal _Ld′/_Ld″$L_d\prime /L_d″$ is a finitely generated K[_Xd]^δ$K{[X_d]}^{\delta }$-module. For small d  , we calculate the Hilbert series of (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$ and find the generators of the K[_Xd]^δ$K{[X_d]}^{\delta }$-module (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$. This gives also an (infinite) set of generators of the algebra (_Ld/_Ld^″)^δ${(L_d/{L_d}^{″})}^{\delta }$.     

Cohen-Macaulay monomial ideals with given radical

We study the set of Cohen-Macaulay monomial ideals with a given radical. Among this set of ideals are the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. It is shown that if there exists one such modification, then there exist indeed infinitely many.     

An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution components.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Rational power series, sequential codes and periodicity of sequences

Let R be a commutative ring. A power series f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z_m for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in F[[x₁,…,x_n]]. We extend Kronecker’s criterion for rationality in F[[x]] to F[[x₁,…,x_n]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F_p for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction.