MOTIVATIONDuring evolution, functional regions in genomic sequences tend to be more highly conserved than randomly mutating 'junk DNA' so local sequence similarity often indicates biological functionality. This fact can be used to identify functional elements in large eukaryotic DNA sequences by cross-species sequence comparison. In recent years, several gene-prediction methods have been proposed that work by comparing anonymous genomic sequences, for example from human and mouse. The main advantage of these methods is that they are based on simple and generally applicable measures of (local) sequence similarity; unlike standard gene-finding approaches they do not depend on species-specific training data or on the presence of cognate genes in data bases. As all comparative sequence-analysis methods, the new comparative gene-finding approaches critically rely on the quality of the underlying sequence alignments.RESULTSHerein, we describe a new implementation of the sequence-alignment program DIALIGN that has been developed for alignment of large genomic sequences. We compare our method to the alignment programs PipMaker, WABA and BLAST and we show that local similarities identified by these programs are highly correlated to protein-coding regions. In our test runs, PipMaker was the most sensitive method while DIALIGN was most specific.AVAILABILITYThe program is downloadable from the DIALIGN home page at http://bibiserv.techfak.uni-bielefeld.de/dialign/.

Novel synthesis strategies based on the geometrical and topological principles outlined here open up pathways to a new class of spherical clusters with icosahedral symmetry of the type (pentagon)12(linker)30—also called keplerates—where the centers of the 12 pentagons span an icosahedron and the centers of the 30 linkers an icosidodecahedron. Remarkably, sizing of a spherical molecule is possible for the first time. In addition to their large size of several nanometers, these molecules show unusually high symmetries. When large numbers of paramagnetic metal centers like 30 FeIII or 20 VO2+ are integrated within their structure, extraordinary spin topologies can be realized on a discrete molecular level. Further functionalization of these systems allows, e.g. to link them forming chains or layers in solid state reactions at room temperature.

363 Ragan (1992), Purvis (1995a), and Sanderson et al. (1998). The purpose of this note is to point out that inherent limitations exist as to what any supertree or consensus tree method can possibly achieve. More precisely, we demonstrate here by simple examples that, although one can easily list various desirable properties that one would like such a method to possess, no method can possess all of these simultaneously. Although this may be viewed as a somewhat negative result, its positive spin-off is that it focuses attention on those types of supertree constructions one should aim for and indicates promising lines for future work. Although we insist that any supertree method must return a single tree, we will briefly discuss extensions that allow several trees to be in the output. One inherent limitation on the solution to the consensus tree problem for unrooted trees has been described by McMorris (1985), who discussed three conditions that cannot simultaneously be satisfied by any consensus method. The conditions we consider here are quite different and somewhat simpler. We will also see that there is a fundamental distinction between rooted and unrooted trees.

The amalgamation of leaf-labeled trees into a single (super)tree that “displays” each of the input trees is an important problem in classification. We discuss various approaches to this problem and show that a simple and well-known polynomial-time algorithm can be used to solve this problem whenever the input set of trees contains a minimum size subset that uniquely determines the supertree. Our results exploit a recently established combinatorial property concerning the structure of such collections of trees.

In this paper, we give a very efficient and completely reliable method to enumerate all combinatorial possibilities for fullerene structures. The method is based on a top-downdivide and conquerapproach and can easily be generalized also for other kinds of spherical structures. A computer program based on this approach was used to produce complete lists of fullerene structures with up to 162 atoms.

A well known result from cluster theory states that there is a 1-to-1 correspondence between dated, compact, rooted trees and ultrametrics. In this paper, we generalize this result yielding a canonical 1-to-1 correspondence between symbolically dated trees and symbolic ultrametrics, using an arbitrary set as the set of (possible) dates or values. It turns out that a rather unexpected new condition is needed to properly define symbolic ultrametrics so that the above correspondence holds. In the second part of the paper, we use our main result to derive, as a corollary, a theorem by H. J. Bandelt and M. A. Steel regarding a canonical 1-to-1 correspondence between additive trees and metrics satisfying the 4-point condition, both taking their values in abelian monoids.

MOTIVATION:DCA is a new computer program for multiple sequence alignment which utilizes a 'divide-and-conquer' type of heuristic approach.AVAILABILITY:The algorithm is freely available from http://bibiserv.TechFak.Uni-Bielefeld.DE/dca/.

We consider the problem of multiple sequence alignment: given Ic sequences of length at most n and a certain scoring function, find an alignment that minimizes the corresponding ''sum of pairs'' distance score.We generalize the divide-and-conquer technique described in [1,2], and present new ideas on how to use efficient search strategies for saving computer memory and accelerating the procedure for three or more sequences. Resulting running times and memory usage are shown for several test cases.

In this note, it is shown that there exists a natural metric on the set[formula]of bounded subsets of R containing their infimum which endowsTwith the structure of an R-tree so that, for everyt∈T, the “number” of connected components ofT\{t} coincides with the cardinality #P(R) of the set of subsets of R. In addition, the set of ends ofTis explicitly determined, and various further features ofTare discussed, too.

We have developed a fast heuristic algorithm for multiple sequence alignment which provides near-to-optimal results for sufficiently homologous sequences. The algorithm makes use of the standard dynamic programming procedure by applying it to all pairs of sequences. The resulting score matrices for pair-wise alignment give rise to secondary matrices containing the additional charges imposed by forcing the alignment path to run through a particular vertex. Such a constraint corresponds to slicing the sequences at the positions defining that vertex, and aligning the remaining pairs of prefix and suffix sequences separately. From these secondary matrices, one can compute - for any given family of sequences - suitable positions for cutting all of these sequences simultaneously, thus reducing the problem of aligning a family of n sequences of average length l in a Divide and Conquer fashion to aligning two families of n sequences of approximately half that length. In this paper, we explain the method for the case of 3 sequences in detail, and we demonstrate its potential and its limits by discussing its behaviour for several test families. A generalization for aligning more than 3 sequences is lined out, and some actual alignments constructed by our algorithm for various user-defined parameters are presented.

Using the Wedderburn-Artin theory of simple artinian rings, Grothendieck′s version of the basic structure theorem of noncommutative Galois theory is derived as a simple corollary of Dedekind′s Lemma.

Given a set function, that is, a map ƒ: P(E) → R ≔ R ∪ {−∞} from the set P(E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x0 ≔ ∅, x1, . . ., xN ≔ E (N ≔ #E) of subsets with max(ƒ) ∈ {ƒ(x0), ƒ(x1), . . ., ƒ(xN)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) : P(E) → R : x ↦ ƒ(x) + Σe ∈ xη(e), where η runs through all real valued weighting schemes η : E → R, in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E → R and every ƒeta-greedy sequence x0, x1, . . ., xN as above, one has max(ƒη) = ƒη(xi) for the unique i ∈ {0, . . ., N} with ƒη(x0) < ƒη(x1) < ··· < ƒη(xi) ≥ ƒη(xi + 1).

In the present paper we prove an identity concerning Pfaffians similar to the well-known Grassmann-Plucker relations for determinants, using tools from multilinear algebra. More precisely, we shall derive the identity as a corollary to an equation concerning skew-symmetric bilinear forms and operators, acting on the exterior algebra Λ(M) of an R-module M over some fixed commutative ring R with 1 ∈ R.

Given a finite set E of cardinality, say, N and a map ƒ:P(E) →R ≔ R∪{−∞}, we define a sequence e1, …, eN of elements of E to be ƒ-greedy if ei ∉ {e1, …, ei−1} and ƒ({e1, …, ei}) ≥ ƒ({e1, …, ei−1} ∪ {e}) for all i ∈ {1, …, N} and all e ∈ E⧸{e1, …, ei−1}. In addition, for any map η : E →R we put ƒη(x) ≔ ƒ(x) + Σe∈xη(e) for all x ⊆ E, and we define ƒ to be a well-layered map if for every η : E →R, every ƒη-greedy sequence e1, …, eN ∈ E and every layer Pi(E) ≔ {y ⊆ E | #y = i} (i = 0, …, N) we have ƒη({e1, …, ei}) = max{ƒη(y) | y ∈ Pi(E)}. In this note, we characterize well-layered maps by some appropriate quantified version of the greedoid exchange condition or—equivalently—by some appropriate greedoidal version of the quantitative relations defining valuated matroids and valuated Δ-matroids, respectively.

In this note-following the line of thought introduced into Galois theory by Emil Artin in (or before?) 1948-a complete proof of the basic structure theorem of that theory is established in altogether less than 50 lines of text (including formulae), using nothing but Dedekind′s lemma, one of its elementary and well-known consequences, and two of the most basic facts of G-set theory. In the remaining pages, the same direct approach is used to establish for a finite group G of automorphisms of a field L with fixed field K the canonical (anti-)equivalence of the category of finite-dimensional L-split K-algebras and the category of finite G-sets as well as the basic existence theorem of Galois theory, that is, the fact that the number of K-algebra automorphisms of a field extension L = K(α1,...,αn) with #(α1,...,αn) = n and ∏ni<=1(X−αi) ∈ K[X] equals the degree (L : K)= DimKL of this extension and is bounded from above by n!.

Cluster analysis deals with procedures which - given a finite collection X of objects together with some kind of local dissimilarity information - identify those subcollections C of objects from X, called clusters, which exhibit a comparatively low degree of internal dissimilarity. In this note we study arbitrary mappings phi which assign to each subcollection A subset of or equal to X of objects its internal degree of dissimilarity phi(A), subject only to the natural condition that A subset of or equal to B subset of or equal to X implies phi(A)less than or equal to phi(B), and we analyse on a rather abstract, purely order theoretic level how assumptions concerning the way such a mapping cp might be constructed from local data (that is, data involving only a few objects at a time) influence the degree of overlapping observed within the resulting family of clusters, - and vice versa.Hence, unlike previous order theoretic approaches to cluster analysis, we do not restrict our attention to nonoverlapping, hierarchical clustering. Instead, we regard a dissimilarity function phi as an arbitrary isotone mapping from a finite partially ordered set Q - e.g. the set P(X) of all subsets A of a finite set X - into a (partially) ordered set R - e.g. the nonnegative real numbers - and we study the correspondence between the two subsets C(phi) and D(phi) of Q, formed by the elements whose images are inaccessible from above and from below, respectively. While D(phi) constitutes the local data structure from which phi can be built up, C(phi) embodies the family of clusters associated with phi.Our results imply that in case Q:=P(X) and R:=R(greater than or equal to 0) one has #D less than or equal to n for all D is an element of D(phi) and some fixed n is an element of N if and only if[GRAPHICS]for all C-0,..., C-n is an element of C(phi) if and only if this holds for all subsets C-0,..., C-n subset of or equal to X, generalizing a well-known criterion for n-conformity of hypergraphs as well as corresponding results due to Batbedat, dealing with the case n=2.

Matroids with an arbitrary domain of coefficients have been introduced in [A. W. M. Dress, Adv. Math.59 (1986), 97–123] and, since, studied in [A. W. M. Dress and W. Wenzel, Adv. Math.77 (1989), 1–36; Adv. Math.86 (1991), 68–110; Bayreuth. Math. Schr.26 (1988), 37–98; Geom. Dedicata34 (1990), 161–197; Appl. Math. Lett.3, No. 2 (1990), 33–35; Adv. Math., in press; M. Wagowski, European J. Combin.10 (1989), 393–398; W. Wenzel, Adv. Math.77 (1989), 37–75; J. Combin. Theory Ser. A57 (1991), 15–45]. In the present paper we study such matroids whose coefficients belong to a particular, but rather natural, class of such domains, the so-called perfect fuzzy rings. These include matroids representable over a ring as well as ordinary, oriented, and valuated matroids. A number of well-known and important results which are known to hold for such matroids (e.g., Tutte's representability theorem and the corresponding results for oriented and valuated matroids), but do not hold for arbitrary matroids with coefficients, as well as some additional results concerning, e.g., (fuzzy) determinant identities, which were not known even in the case of oriented matroids, are shown to hold more generally for matroids with coefficients in perfect domains.