(2) If K ∈ B(X) is compact, then for all λ ∈ C \ {0}, K − λ1 is Fredholm with index zero. (3) The shift operator S± ∈ B(`p) for 1 ≤ p ≤ ∞ defined by (S±x)n = xn±1 is Fredholm with index ±1. (4) If X,Y are finite dimensional and T ∈ B(X,Y ), then by the Rank-Nullity Theorem, ind(T ) = dim(X)− dim(Y ). Lemma 3. Suppose E,F ⊆ X are closed subspaces with F finite dimensional. (1) The subspace E + F...

We revisit the kink-like parametrization of the deceleration parameter q(z) [1], which considers a transition, at redshift zt, from cosmic deceleration to acceleration. In this parametrization the initial, at z zt, value of the q-parameter is qi, its final, z = −1, value is qf and the duration of the transition is parametrized by τ . By assuming a flat space geometry we obtain constraints on th...

In the original formulation of the problem, the Budach mouse is a kind of finite automaton with a very simple program, a cycle of states, q0 d1 −→ q1 d2 −→ . . . qn−1 dn −→ q0, with d1, . . . , dn ∈ {N,E}. The mouse moves among integer points in the octant of the Cartesian plane delimited by the lines y = 0 and y = x. It begins on (1, 0), at state q0, and advances in steps which increase the st...

Underlying cosmologies with deceleration parameter, q0 = −0.51 +0.03 −0.24, may be interpreted as having q0 = −0.55, on the basis of weak gravitational lensing effects. 1. Computing the Three-Dimensional Shear The algorithm used for determining the three-dimensional shear has been described fully by Couchman, Barber & Thomas (1998). Of particular interest is the variable softening feature in th...

The Malmquist bias in luminosity distances for gaussian standard candles is discussed within cosmological models where the Euclidean r3-law for volumes and r−2-law for fluxes is not valid. Furthermore, the influence of K-corrections and luminosity evolution are analyzed. It is noted that the usual way of comparing theoretical predictions and data points in the Hubble diagram (log z vs. m) shoul...

In [2] it has been shown that infinitary strong normalization (SN∞) is Π11-complete. Suprisingly, it turns out that infinitary weak normalization (WN∞) is a harder problem, being Π12-complete, and thereby strictly higher in the analytical hierarchy. We assume familiarity with infinitary term rewriting; we further reading we refer to [5,4]. 1 Infinitary Strong Normalization and Reachability Defi...

Ti50Pd50−xCrx is a high-temperature shape-memory alloy with a martensitic transformation temperature strongly dependent on the Cr composition. Prior to the transformation, a premartensitic phase is present with an incommensurate modulated cubic lattice with wave vector of q0=(0.22,0.22,0). The temperature dependence of the diffuse scattering in the cubic phase is measured as a function temperat...

The case is simple: there is no compelling theoretical argument against a cosmological constant and ΛCDM is the only CDM model that is consistent with all present observations. ΛCDM has two noteworthy features: it can be falsified in the near future (the prediction q0 ∼ − 1 2 is an especially good test), and, if correct, it has important implications for fundamental physics.

A new cosmological model provides solutions for the flatness, smoothness, monopole, and classical singularity problems, with no adjustment of parameters. The prediction of a scale factor R (t) which is proportional to t (rather than t) yields a comfortably large age for the universe, and also implies that the deceleration parameter q0 is exactly zero.