Classification of Graeco‐Latin Cubes

A q‐ary code of length n, size M, and minimum distance d is called an code. An code with is said to be maximum distance separable (MDS). Here one‐error‐correcting () MDS codes are classified for small alphabets. In particular, it is shown that there are unique (5, 5³, 3)₅ and (5, 7³, 3)₇ codes and equivalence classes of (5, 8³, 3)₈ codes. The codes are equivalent to certain pairs of mutually orthogonal Latin cubes of order q, called Graeco‐Latin cubes.     

Enumerating Perfect Matchings in n-Cubes

The perfect matchings in the n-cube have earlier been enumerated for n?≤?6. A dynamic programming approach is here used to obtain the total number of perfect matchings in the 7-cube, which is 391 689 748 492 473 664 721 077 609 089. The number of equivalence classes of perfect matchings is further shown to be 336 in the 5-cube, 356 788 059 in the 6-cube and 607 158 046 495 120 886 820 621 in the 7-cube. The techniques used can be generalized to arbitrary bipartite and general graphs.     

Quarter pinched homogeneous spaces of negative curvature

Certain properties of simply connected spaces with quarter pinched sectional curvature K, −4<K≤1, are considered. Additional conditions guaranteeing that the corresponding group of isometries is a symmetric space are given. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 39–40.     

Prearranged Glycosides,Part 10. Intramolecular Glycosylation with Cellobiosyl,Lactosyl, and Maltosyl Donors

ABSTRACT

Acetyl protected 1,2-O-(1-methoxyethylidene)-disaccharides 1 of maltose, cellobiose, and lactose, respectively were converted via the corresponding benzyl protected couterparts 2, the benzyl protected phenyl 2-O-acetyl- 3 and 2-O-unprotected 1-thio-glycoside disaccharides 4 into 2-O-succinoylated disaccharides 5. The latter were esterified with benzyl 2-O-benzoyl-4,6-di-O-benzylidene-α-D-glucopyranoside (6) to afford succinyl linked derivatives 7 the benzylidene groups of which were regioselectively opened to give prearranged glycoside trisaccharides 8. Intramolecular glycosylation of the latter with N-iodosuccinimide resulted in exclusive formation of the corresponding α-(1→4)-linked trisaccharides 9. No influence of the donor moiety on the diastereoselectivity of the intramolecular glycosylation was observed.     

The first,the biggest,and other such considerations

We investigate the relation between the underlying dynamics of randomly evolv ing systems and the extrema statistics for such systems. Independent processes, Fokker-Planck processes and Lévy processes are considered.     

There exists no (15,5,4) RBIBD

An (n, M, d)_q code is a q‐ary code of length n, cardinality M, and minimum distance d. We show that there exists no (15,5,4) resolvable balanced incomplete block design (RBIBD) by showing that there exists no (equidistant) (14,15,10)₃ code. This is accomplished by an exhaustive computer search using an orderly algorithm combined with a maximum clique algorithm. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 227–232, 2001     

A Note on Toeplitz’ Conjecture

In 1911, Toeplitz made a conjecture asserting that every Jordan curve in $\mathbb{R}^{2}$ contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n×n grid is at least $1/\sqrt{2}$ times the side length of the largest axis-aligned square contained inside the curve. Conjecture C implies Toeplitz’ conjecture and is verified computationally for n≤13.     

On the non-equivalence of two standard random walks

We focus on two models of nearest-neighbour random walks on d$d$-dimensional regular hyper-cubic lattices that are usually assumed to be identical—the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll–Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1$1$. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.