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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.
On derived varieties
(1996)
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.
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.
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 \(L_k\).
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.
Software development organizations measure their real-world processes, products, and resources to achieve the goal of improving their practices. Accurate and useful measurement relies on explicit models of the real-world processes, products, and resources. These explicit models assist with planning measurement, interpreting data, and assisting developers with their work. However, little work has been done on the joint use of measurem(int and process technologies. We hypothesize that it is possible to integrate measurement and process technologies in a way that supports automation of measurement-based feedback. Automated support for measurementbased feedback means that software developers and maintainers are provided with on-line, detailed information about their work. This type of automated support is expected to help software professionals gain intellectual control over their software projects. The dissertation offers three major contributions. First, an integrated measurement and
process modeling framework was constructed. This framework establishes the necessary foundation for integrating measurement and process technologies in a way that will permit automation. Second, a process-centered software engineering environment was developed to support measurement-based feedback. This system provides personnel with information about the tasks expected of them based on an integrated set of measurement and process views. Third, a set of assumptions and requirements about that system were examined in a controlled experiment. The experiment compared the use of different levels of automation to evaluate the acceptance and effectiveness of measurement-based feedback.
The calculation of form factors is an important problem in computing the global illumination in the radiosity setting. Closed form solutions often are only available for objects without obstruction and are very hard to calculate. Using Monte Carlo integration and ray tracing provides a fast and elegant tool for the estimation of the form factors. In this paper we show, that using deterministic low discrepancy sample points is superior to random sampling, resulting in an acceleration of more than half an order of magnitude.
Quasi-Monte Carlo Radiosity
(1996)
The problem of global illumination in computer graphics is described by a second kind Fredholm integral equation. Due to the complexity of this equation, Monte Carlo methods provide an interesting tool for approximating
solutions to this transport equation. For the case of the radiosity equation, we present the deterministic method of quasi-rondom walks. This method very efficiently uses low discrepancy sequences for integrating the Neumann series and consistently outperforms stochastic techniques. The method of quasi-random walks also is applicable to transport problems in settings other
than computer graphics.