finite dimensional backward shift invariant subspaces of arveson spaces a description is given of finite dimensional backward shift invariant subspaces of arveson spaces which are certain multivariable analogues of the hardy space of the unit disk

structural properties of context free sets of graphs generated by vertex replacement abstract we establish that a vertex replacement set of graphs i e a set of graphs generated by a c ednce or equivalently by a separated handle rewriting graph grammar is hyperedge replacement i e is generated by a hyperedge replacement graph grammar iff its graphs do not contain arbitrary large complete bipartite graphs kn n as subgraphs another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices these properties are decidable by means of an appropriate extension of the theorem by parikh that characterizes the commutative images of context free languages we extend these results to hypergraphs

the schur algorithm and reproducing kernel hilbert spaces in the ball abstract using reproducing kernel hilbert spaces methods we develop a schur type algorithm for a subclass of the functions analytic and contractive in the ball we also consider the nevanlinna pick interpolation problem in that class

backward shift operator and finite dimensional de branges rovnyak spaces in the ball recently bolotnikov and rodman characterized finite dimensional backward shift invariant subspaces of the arveson space in this note we study finite dimensional backward shift invariant hilbert spaces contractively included in the arveson space to that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions

boundary interpolation in the ball we solve a boundary interpolation problem in the reproducing kernel hilbert space of functions analytic in the unit ball of cn with reproducing kernel 1 1 1nzkwk we introduce the notion of brune factor or blaschke potapov factor of the third kind in this setting

the monadic second order logic of graphs xiv uniformly sparse graphs and edge set quantifications we consider the class usk of uniformly k sparse simple graphs i e the class of finite or countable simple graphs every finite subgraph of which has a number of edges bounded by k times the number of vertices we prove that for each k every monadic second order formula intended to express a graph property that uses variables denoting sets of edges can be effectively translated into a monadic second order formula where all set variables denote sets of vertices and that expresses the same property of the graphs in usk this result extends to the class of uniformly k sparse simple hypergraphs of rank at most m for any k and m it follows that every subclass of usk consisting of finite graphs of bounded clique width has bounded tree width clique width is a graph complexity measure similar to tree width and relevant to the construction of polynomial algorithms for np complete problems on special classes of graphs

trees grids and mso decidability from graphs to matroids monadic second order mso logic has proved to be a useful tool in many areas of application reaching flom decidability and complexity to picture processing correctness of programs and parallel processes to characterize the structural borderline between decidability and undecidability is a classical research problem here this problem is related to questions in computational complexity especially to the model checking problem for which many tools developed in the area of decidability have proved to be useful for more than two decades it was conjectured in d seese the structure of the models of decidable monadic theories of graphs ann pure appl logic 53 1991 169 195 that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees it is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field matroids can be viewed as a wide generalization of graphs and they seem to capture some second order properties in a more suitable way than graphs themselves cf the recent development in matroid structure theory j f geelen a h m gerards g p whittle branch width and well quasi ordering in matroids and graphs j combin theory ser b 84 2002 270 290 j f geelen a h m gerards n robertson g p whittle excluding a planar graph from a gf q representable matroid manuscript 2003 more exactly we prove for every finite field f that any class of f representable matroids with a decidable mso theory must have uniformly bounded branch width moreover we show that bounding the branch width of all matroids in general is not sufficient to obtain a decidable mso theory our paper gives a rather detailed introduction to these different subjects and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms and can shed light on some old open problems

the monadic second order logic of graphs x linear orderings graphs are finite and handled as relational structures we give some answers to the following general questions 1 1 for which classes of graphs c is it possible to specify a linear ordering of the set of vertices of each graph of c by fixed monadic second order formulas 2 2 for which classes of graphs c does there exist an extension l of monadic second order logic such that a subclass l of c is recognizable if and only if it is the class of graphs in c that satisfy a formula of l in this paper recognizability is understood in an algebraic sense relative to a finite set of graph operations and basic graphs that generate all graphs of c 3 3 for which classes of graphs c is it possible to construct in every graph of the class and by fixed formulas of a suitable extension of monadic second order logic its hierarchichal structure i e a finite term written with the operations and basic graphs of 2 that defines the considered graph applications concern dependency graphs of partially commutative words partial k paths cographs and graphs the modular decomposition of which uses prime graphs of bounded size

a parametrized algorithm for matroid branch width branch width is a structural parameter very closely related to tree width but branch width has an immediate generalization from graphs to matroids we present an algorithm that for a given matroid m of bounded branch width t which is represented over a finite field finds a branch decomposition of m of width at most 3t in cubic time then we show that the branch width of m is a uniformly fixed parameter tractable problem other applications include recognition of matroid properties definable in the monadic second order logic for bounded branch width and s i oum approximating rank width and clique width quickly in proceedings of the 31st international workshop on graph theoretic concepts in computer science springer verlag heidelberg to appear a cubic time approximation algorithm for graph rank width and clique width

caratheodory fejer interpolation in the ball with mixed derivatives we use reproducing kernel hilbert space methods to solve a caratheodory fejer type interpolation problem with mixed derivatives corresponding to a so called lower inclusive set in the class of schur multipliers

rabin s theorem in the concurrency setting a conjecture rabin s theorem says that the monadic second order theory of the infinite binary tree is decidable this result has had a far reaching influence in the theory of branching time temporal logics a simple consequence of rabin s theorem is that for every finite state transition system the monadic second order theory of its computation tree is decidable concurrency theory strongly suggests that finite 1 safe petri nets or simply net systems are a natural generalization of the notion of a finite state transition system while labelled event structures arising as the unfoldings of net systems are the proper counterparts to the computation trees obtained by unwinding finite sequential transition systems it is easy to define the monadic second order theory of such event structures it turns out that unlike the sequential case not every net system i e its event structure unfolding has a decidable monadic second order theory this gives rise to the question which net systems admit a decidable monadic second order theory here we present a conjecture based on a property called grid freeness our conjecture is that a net system has a decidable monadic second order theory iff its event structure unfolding is grid free we show that it is decidable whether a net system has this property we also prove that the monadic second order theory of a net system is undecidable if its event structure unfolding is not grid free in addition we show that our conjecture can be effectively reduced to the sub class of free choice net systems finally we point out how the positive resolution of our conjecture will settle the decidability of a range of distributed controller synthesis problems

recursive t matrix methods for scattering from multiple dielectric and metallic objects we present an efficient stable recursive t matrix algorithm to calculate the scattered field from a heterogeneous collection of spatially separated objects the algorithm is based on the use of higher order multipole expansions than those typically employed in recursive t matrix techniques the use of these expansions introduces instability in the recursions developed by chew 1990 and by wang and chew 1990 specifically in the case of near field computations by modifying the original recursive algorithm to avoid these instabilities we arrive at a flexible and efficient forward solver appropriate for a variety of scattering calculations the algorithm can be applied when the objects are dielectric metallic or a mixture of both we verify this method for cases where the scatterers are electrically small fraction of a wavelength or relatively large 12 spl lambda while developed for near field calculation this approach is applicable for far field problems as well finally we demonstrate that the computational complexity of this approach compares favorably with comparable recursive algorithms

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

the monadic second order logic of graphs vii graphs as relational structures abstract relational structures form a unique framework in which various types of graphs and hypergraphs can be formalized and studied we define operations on structures that are compatible with monadic second order logic and that are powerful enough to represent context free graph and hypergraph grammars of various types namely the so called hyperedge replacement c ednce and separated handle rewriting grammars several results concerning monadic second order properties of the generated sets are obtained in a uniform way we also give a logical characterization of the equational sets of structures that generalizes the ones obtained by engelfriet and courcelle for hyperedge replacement and c ednce sets of graphs

the curvature of a hilbert module over ℂ z1 zd a notion of curvature is introduced in multivariable operator theory the curvature invariant of a hilbert module over z1 zd is a nonnegative real number which has significant extremal properties which tends to be an integer and which is hard to compute directly it is shown that for graded hilbert modules the curvature agrees with the euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring this result is a higher dimensional operator theoretic counterpart of the gauss bonnet formula which expresses the average gaussian curvature of a compact oriented riemann surface as the alternating sum of the betti numbers of the surface and it solves the problem of calculating the curvature of graded hilbert modules the proof of that result is based on an asymptotic formula which expresses the curvature of a hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the euler characteristic

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

rational hyperholomorphic functions in r4 we introduce the notion of rationality for hyperholomorphic functions functions in the kernel of the cauchy fueter operator following the case of one complex variable we give three equivalent definitions the first in terms of cauchy kovalevskaya quotients of polynomials the second in terms of realizations and the third in terms of backward shift invariance also introduced and studied are the counterparts of the arveson space and blaschke factors

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

the monadic second order logic of graphs xii planar graphs and planar maps we prove that we can specify by formulas of monadic second order logic the unique planar embedding of a 3 connected planar graph if the planar graph is not 3 connected but given with a linear order of its set of edges we can also define a planar embedding by monadic second order formulas we cannot do so in general without the ordering even for 2 connected planar graphs the planar embedding of a graph can be specified by a relational structure called a map which is a graph enriched with a circular ordering of the edges incident with each vertex this circular ordering called a rotation system represents a planar embedding of the neighbourhood of each vertex for each connected map one can define a linear order on its vertices by formulas of monadic second order logic hence we have for planar graphs some kind of equivalence between linear orderings and planar embeddings

a recursive single source surface integral equation analysis for wave scattering by heterogeneous dielectric bodies the problem of electromagnetic wave scattering by heterogeneous dielectric bodies is formulated in a recursive manner by organizing their homogeneous subregions into hierarchical multiply nested structures the inner details of each multiply nested body are completely accounted for by an equivalent surface representation that yields the electric and magnetic fields tangent to the body only in terms of a single unknown electric surface current density distributed on its outer surface in this manner the problem of wave scattering by heterogeneous dielectric bodies is reduced to a scattering problem over their outermost surfaces in terms of only a single unknown current density for a large number n of different homogeneous dielectric subregions within such a heterogeneous body the proposed method has a computational complexity of o n sup 1 5 and storage requirements that increase in proportion to o n furthermore the equivalent surface representation derived for a particular subregion is invariant under rotation and translation and may therefore be applied to identical subregions without repeating the computation the fields at any interior points are calculated by a fast backward recursion

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

the monadic second order logic of graphs xiii graph drawings with edge crossings we introduce finite relational structures called sketches that represent edge crossings in drawings of finite graphs we consider the problem of characterizing sketches in monadic second order logic we answer positively the question for framed sketches i e for those representing drawings of graphs consisting of a planar connected spanning subgraph the frame augmented with additional edges that may cross one another and that may cross the edges of the frame we prove the 3 edge theorem stating that a structure of appropriate type with a frame is a sketch if and only if every induced substructure representing the frame and at most 3 edges not in the frame is a sketch

invariant subspaces and nevanlinna pick kernels abstract a theorem of beurling lax halmos represents a subspace m of h2 c d the hardy space of analytic functions with values in the hilbert space e and square summable power series invariant for multiplication by z as h2 f where f is a subspace of e and is an inner function with values in l f e when the hardy space is replaced by the hilbert space h k determined by a nevanlinna pick kernel k such as the dirichlet kernel or the row contraction kernel on the ball in c d the blh theorem survives with f an auxiliary hilbert space and a l f e valued function which is inner in the sense that the operator m of multiplication by is a partial isometry under mild additional hypotheses when e c mz the operator of multiplication by z is cellularly indecomposable and has the codimension one property however if m is invariant for mz m mz m need not be a cyclic subspace for mz restricted to m

complementation in krein spaces a generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product complementation is a construction in the geometry of hilbert space which was applied to the invariant subspace theory of contractive transformations in hilbert space by james rovnyak and the author 6 the concept was later formalized by the author 3 continuous and contractive transformations in krein spaces appear in the estimation theory of riemann mapping functions 4 it is therefore of interest to know whether a generalization of complementation theory applies in krein spaces such a generalization is now obtained the results are also of interest in the invariant subspace theory of continuous and contractive transformations in krein spaces 5 the vector spaces considered are taken over the complex numbers a scalar product for a vector space 1 is a complex valued function a b of a and b in 1 which is linear symmetric and nondegenerate linearity means that the identity aa 3b c a a c 3 b c holds for all elements a b and c of 1 when a and are complex numbers symmetry means that the identity b a a b holds for all elements a and b of 1 nondegeneracy means that an element a of 1 is zero if the scalar product a b is zero for every element b of 1 every element b of 1 determines a linear functional bon 1 which is defined by b a a b for every element a of 1 the weak topology of 1 is the weakest topology with respect to which bis a continuous linear functional on 1 for every element b of 1 the weak topology of 1 is a locally convex topology having the property that every continuous linear functional on 1 is of the form bfor an element b of 1 the element b is then unique the antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product a fundamental example of a vector space with scalar product is a hilbert space a krein space is a vector space with scalar product which is the orthogonal sum of a hilbert space and the antispace of a hilbert space received by the editors october 10 1986 and in revised form february 23 1987 the results of the paper were presented to the department of mathematics indiana and purdue university in indianapolis on march 27 1987 as the ernest j johnson colloquium 1980 mathematics subject classification 1985 revision primary 46d05 research supported by the national science foundation 1988 american mathematical society 0002 9947 88 1 00 25 per page

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems