The massive scalar field in a closed Friedmann universe model - new rigorous results
For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove D ≥ 1, where D ≥ is the dimension of the set of solutions which can be integrated up to t ∞ (D > 0 was conjectured by PAGE (1984)). We discuss concepts like “the probability of the appearance of a sufficiently long inflationary phase” and argue that it is primarily a probability measure μ in the space V of solutions (and not in the space of initial conditions) which has to be applied. μ is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of μ can be given. 
 
 
 
We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented. 
 
 
 
Fur das minimal gekoppelte Skalarfeld in Einsteinscher Gravitationstheorie betrachten wir den Losungsraum in der Klasse der geschlossenen Friedmannmodelle des Universums. Wir beweisen, das D ≥ 1 gilt, wobei D die Dimension der Menge derjenigen Losungen ist, die sich bis t ∞ integrieren lassen. (D > 0 war von PAGE (1984) vermutet worden.) Wir diskutieren Begriffe wie, ”die Wahrscheinlichkeit des Auftretens einer hinreichend langen inflationaren Phase„ und argumentieren, das man primar ein Wahrscheinlichkeitsmas μ im Raum der Losungen V (und nicht im Raum der Anfangswerte) anwenden mus. Fur kosmologische Modelle vom Bianchi-Typ I gibt es eine naturliche Definition fur μ, da dort V ein kompakter Wurfel ist. Die Probleme mit dem geschlossenen Friedmannmodell (die zu Kontroversen in der Literatur fuhrten) ergeben sich aus der Tatsache, das V dort eine komplizierte Geroch-Topologie hat, die weder kompakt noch hausdorffsch ist, so das sich keine naturliche Definition fur μ angeben last. 
 
 
 
Wir schliesen: Zwar last sich der heutige Zustand unseres Universums durch Modelle des hier diskutierten Typs erklaren, jedoch last sich dabei das anthropische Prinzip nicht vollig umgehen.

title: Inhomogeneous Cosmological Models with Flat Slices Generated from the EINSTEIN‐DE SITTER Universe
abstract: A family of cosmological models in considered which in a certain synchronized system of reference possess flat slices t = const. They are generated from the EINSTEIN-DE SITTER universe by a suitable transformation. Under physically reasonable presumptions these transformed models fulfil certain energy conditions. 
 
 
 
Es wird eine Familie kosmologischer Modelle betrachtet, die in einemi gewissen synchronisierten Bezugssystem flache Schichten t = const. besitzen. Sie werden mittels einer geeigneten Transformation aus dem EINSTEIN-DE SITTER-Universum erzeugt. Diese transformierten Modelle erfullen unter physikalisch sinnvollen Voraussetzungen gewisse Energie bedingungen.

title: New exact solutions for power‐law inflation Friedmann models
abstract: We consider the spatially flat Friedmann model 
 
 
 
For a ≈ tp, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = Rm with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write 
 
 
 
The general solutions reads Q(a) = (ab + C)f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.

title: 4D wormhole with signature change in the presence of extra dimensions
abstract: A regular vacuum solution in 5D gravity on the principal bundle with the U(1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lorentzian (Euclidean) spacetimes connected by a Euclidean (Lorentzian) throat. The 4D Lorentzian observer sees these two null hypersurfaces as electric charges.

title: The Newtonian limit of fourth-order gravity
abstract: The weak-field slow-motion limit of fourth-order gravity will be discussed. Let us consider the gravitational theory defined by the Lagrangian Lg = (8πG) −1 ( R/2 + (αRijR ij + βR)l ) . (1) G is Newton’s constant, l a coupling length and α and β numerical parameters. Rij and R are the Ricci tensor and its trace. Introducing the matter Lagrangian Lm and varying Lg + Lm one obtains the field equation Eij + αHij + βGij = 8πGTij . (2) For α = β = 0 this reduces to General Relativity Theory. The explicit expressions Hij and Gij can be found in STELLE (1978). In a well-defined sense, the weak-field slow-motion limit of Einstein’s theory is just Newton’s theory, cf. DAUTCOURT (1964). In the following we consider the analogous problem for fourth order gravity eqs. (1), (2). For the special cases α = 0 (PECHLANER, SEXL(1966), POLIJEVKTOVNIKOLADZE (1967)), α + 2β = 0 (HAVAS (1977), JANKIEWICZ (1981)) 1 and α+ 3β = 0 (BORZESZKOWSKI, TREDER, YOURGRAU (1978)) this has already been done in the past. Cf. also ANANDAN (1983), where torsion has been taken into account. The slow-motion limit can be equivalently described as the limit c→ ∞, where c is the velocity of light. In this sense we have to take the limit G→ 0 while G · c and l remain constants. Then the energy-momentum tensor Tij reduces to the rest mass density ρ: Tij = δ 0 i δ 0 j ρ , (3) x = t being the time coordinate. The metric can be written as ds = (1 − 2φ)dt − (1 + 2ψ)(dx + dy + dz) . (4) Now eqs. (3) and (4) will be inserted into eq. (2). In our approach, products and time derivatives of φ and ψ can be neglected, i.e., R = 4∆ψ − 2∆φ , where ∆f = f,xx + f,yy + f,zz . Further R00 = −∆φ, H00 = −2∆R00 − ∆R and G00 = −4∆R, where l = 1. Then it holds: The validity of the 00-component and of the trace of eq. (2), R00 −R/2 + αH00 + βG00 = 8πGρ (5) and − R− 4(α+ 3β)∆R = 8πGρ , (6) imply the validity of the full eq. (2). Now, let us discuss eqs. (5) and (6) in more details: Eq. (5) reads − ∆φ−R/2 + α(2∆∆φ− ∆R) − 4β∆R = 8πGρ . (7)

title: Exact Cosmological Solutions of Nonlinear F(R)-Gravity
abstract: We report on the cited papers refs. 1 - 18 from the following points of view: What do we exactly know about solutions when no exact solution (in the sense of "solution in closed form") is available? In which sense do these solutions possess a singularity? In which cases do conformal relations and/or dimensional reductions simplify the deduction? Furthermore, we outline some open questions worth of being studied in future research.

title: On an algorithm solving two-level programming problems with nonunique lower level solutions
abstract: In the paper, an algorithm is presented for solving two-level programming problems. This algorithm combines a direction finding problem with a regularization of the lower level problem. The upper level objective function is included in the regularzation to yield uniqueness of the follower's solution set. This is possible if the problem functions are convex and the upper level objective function has a positive definite Hessian. The computation of a direction of descent and of the step size is discussed in more detail. Afterwards the convergence proof is given.Last but not least some remarks and examples describing the difficulty of the inclusion of upper-level constraints also depending on the variables of the lower level are added.

title: Models for Constrained Motion and d'Alembert's Principle
abstract: A theorem of Arnol'd and Gallavotti states that by a suitable potential of increasing rigidity a mechanical system with conservative forces can be constrained to a hypersurface of the configuration space and its motion approaches a solution of the reduced Lagrangian equation. In this paper the theorem will be generalized to time- and velocity-dependent forces including time-dependent constraints and proved with the aid of the method of averaging. This technique also allows us to identify the constraint force as the averaged negative gradient of the constraint potential.