Stability of vertices in random boolean cellular automata

Based on computer simulations, Kauffman (Physica D, 10, 145-156, 1984) made several generalizations about a random Boolean cellular automaton which he invented as a model of cellular metabolism. Here we give the first rigorous proofs of two of Kauffman's generalizations: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automaton's behavior is small compared to that of a random mapping with the same number of states; and reversal of the states of a large fraction of the vertices does not affect the cycle to which the automaton moves.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Unusual reactivity of ortho-carbonylphenylboronic acids with diethanolamine

The reaction of carbonyl-substituted phenylboronic acids with diethanolamine has been investigated. For compounds containing carbonyl groups at the meta or para positions the reaction occurs exclusively at the boronic acid. Different reactivities were observed for ortho-substituted acetyl- and formylphenylboronic acids where the reaction occurred at both the boronic acid and at the carbonyl group. X-ray studies revealed the presence of a polycyclic structure with tetravalent boron and nitrogen atoms.     

Synthesis of novel tri-podal mesogenic alkenes and side-chain polysiloxanes

Low molecular weight tri-podal biphenyl- and benzoate-type mesogens [C₆H₅C₆H₄O(CH₂)₅SiMe₂CH₂CH₂SiMe₂]₃CH (4), [C₁₁H₂₃O(C₆H₄)₂O(CH₂)₅SiMe₂]₃CH (5) and [MeOC₆H₄OC(O)C₆H₄O(CH₂)₅SiMe₂]₃CH (6) (C₆H₄ = 1,4-phenylene) were obtained, from branched silyl substituted methane precursors [CH₂CH(Me)₂Si]₃CH (1) and (HMe₂Si)₃CH (2). The biphenyl-containing ones (4) and (5) were converted into terminal alkenes, which were subsequently hydrosilylated with poly(methylsiloxanes). The polymer derived from (5) exhibited mesomorphic properties. Such systems have the potential to significantly increase the density of liquid crystal rod-like structures in side chains of linear polymers (or dendritic liquid crystal polymers).     

Inclusion-dependent mechanism of modification of cyclodextrins with heterocycles

Mono(6-deoxy-dimethylpyridinium)-β-cyclodextrins have been synthesized in reaction of mono (p-toluenesulfonyl) derivative of β cyclodextrin with the appropriate lutidine under microwave irradiation and conventional conditions. The results indicate that the mechanism consists of inclusion complex formation.     

2,3‐Difluoro‐4‐formyl­phenyl­boronic acid

The mol­ecule of the title compound, 2,3‐F₂‐4‐(CHO)C₆H₂B(OH)₂ or C₇H₅BF₂O₃, contains a formyl group coplanar with the benzene ring. The boronic acid group is twisted with respect to the benzene ring plane. The mol­ecules are organized into infinite chains via inter­molecular O—H⋯O hydrogen bonds. These chains are additionally connected via strong O—H⋯O hydrogen bonds, producing a folded layer structure perpendicular to the a axis. These layers are paired due to B⋯F inter­actions.     

Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )

Designs, Codes and Cryptography - The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics...