Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation $$\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $\Lambda_p$ with $\Lambda_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.Comment: 13 pages, 6 figures, submitted to J. Stat. Phy

Operators between subspaces and quotients of L1

We provide an unified approach of results of L. Dor on the complementation of the range, and of D. Alspach on the nearness from isometries, of small into isomorphisms of L1. We introduce the notion of small subspace of L1 and show lifting theorems for operators between quotients of L1 by small subspaces. We construct a subspace of L1 which shows that extension of isometries from subspaces of L1 to the whole space are no longer true for isomorphisms, and that nearly isometric isomorphisms from subspaces of L1 into L1 need not be near from any isometry.Comment: 35 page

Transplanted olfactory ensheathing cells promote regeneration of cut adult rat optic nerve axons

Transplantation of olfactory ensheathing cells into spinal cord lesions promotes regeneration of cut axons into terminal fields and functional recovery. This repair involves the formation of a peripheral nerve-like bridge in which perineurial-like fibroblasts are organized into a longitudinal stack of parallel tubular channels, some of which contain regenerating axons enwrapped by Schwann-like olfactory ensheathing cells. The present study examines whether cut retinal ganglion cell axons will also respond to these cells, and if so, whether they form the same type of arrangement. In adult rats, the optic nerve was completely severed behind the optic disc, and a matrix containing cultured olfactory ensheathing cells was inserted between the proximal and distal stumps. After 6 months, the transplanted cells had migrated for up to 10 mm into the distal stump. Anterograde labeling with cholera toxin B showed that cut retinal ganglion cell axons had regenerated through the transplants, entered the distal stump, and elongated for 10 mm together with the transplanted cells. Electron microscopy showed that a peripheral nerve-like tissue had been formed, similar to that seen in the spinal cord transplants. However, in contrast to the spinal cord, the axons did not reach the terminal fields, but terminated in large vesicle-filled expansions beyond which the distal optic nerve stump was reduced to a densely interwoven mass of astrocytic processes

The Precise Formula in a Sine Function Form of the norm of the Amplitude and the Necessary and Sufficient Phase Condition for Any Quantum Algorithm with Arbitrary Phase Rotations

In this paper we derived the precise formula in a sine function form of the norm of the amplitude in the desired state, and by means of he precise formula we presented the necessary and sufficient phase condition for any quantum algorithm with arbitrary phase rotations. We also showed that the phase condition: identical rotation angles, is a sufficient but not a necessary phase condition.Comment: 16 pages. Modified some English sentences and some proofs. Removed a table. Corrected the formula for kol on page 10. No figure