rigidity and energy suppose one holds two sticks in the form of a cross in one hand and places a rubber band in tension around the four ends when it is released it comes to rest in the shape of a convex quadrilateral in a plane it always returns to the same shape no matter how it is distorted as long as the ends do not slip and the rubber band does not break this is a very simple example of a rigid framework of the type we discuss here energy explains why a framework such as the one above is rigid the rubber band deforms in such a way as to minimize the total energy in the framework only in the final deformed shape will the framework have a minimum energy this idea of introducing energy functions is very useful it can be used to prove a key lemma that was used by cauchy 5 in 1813 to show that convex polyhedral surfaces are rigid it can explain why some and perhaps all of r buckminster fuller s tensegrity structures 10 stay up when applied to spider webs it shows why they can only take on certain geometric shapes it also can be used to prove conjecture 6 of branko gri inbaum and g c shephard in their lectures on lost mathematics 12 this conjecture says that if a framework in the shape of a convex polygon with rods sticks on the boundary and cables inside is rigid in the plane so is the framework obtained by reversing the roles of rods and cables it is interesting to compare some of these results with the opening a rm theorem of axel schur 18 see chern 6 also which is a very close smooth analogue to cauchy s lemma schur s theorem says that if a convex planar smooth arc the arm is opened that is it is moved to another position with the same length but with corresponding points having smaller curvature then the two ends are moved apart another amusing application is to show that a regular pentagon in 4 space a pentagon with all 5 sides equal and all 5 angles between the sides equal has its angles bounded between 36 and 108 a comment of o bottema in 4

on the relation between the positive inertia index and negative inertia index of weighted graphs abstract let g w be an edge weighted graph and let a g w be its adjacency matrix the positive inertia index respectively the negative inertia index of g w denoted by p g w resp p g w is defined to be the number of positive eigenvalues resp negative eigenvalues of a g w clearly p g w p g w is the rank of g w whereas p g w p g w is defined as the signature of g w in this paper some upper and lower bounds on p g w p g w are determined for the weighted chordal graphs resp bipartite graphs complete graphs cactus graphs join graphs bisplit graphs power graphs

the signature of two generalizations of line graphs abstract the signature s g of a graph g is defined as the difference between its positive inertia index and negative inertia index in 2013 h ma et al 1 conjectured that c 3 g s g c 5 g for an arbitrary simple graph g where c 3 g denotes the number of cycles in g of length 3 modulo 4 c 5 g denotes the number of cycles in g of length 1 modulo 4 in this paper we prove that this conjecture holds for claw free graphs and graphs with least eigenvalue 2

on strongly norm attaining lipschitz maps abstract we study the set sna m y of those lipschitz maps from a complete pointed metric space m to a banach space y which strongly attain their lipschitz norm i e the supremum defining the lipschitz norm is a maximum extending previous results we prove that this set is not norm dense when m is a length space or local or when m is a closed subset of r with positive lebesgue measure providing new examples which have very different topological properties than the previously known ones on the other hand we study the linear properties which are sufficient to get lindenstrauss property a for the lipschitz free space f m over m and show that all of them actually provide the norm density of sna m y in the space of all lipschitz maps from m to any banach space y next we prove that sna m r is weakly sequentially dense in the space of all lipschitz functions for all metric spaces m finally we show that the norm of the bidual space of f m is octahedral provided the metric space m is discrete but not uniformly discrete or m is infinite

a parametric family of correlation structures for the analysis of longitudinal data in epidemiological settings we are often faced with numerous short time series and a parsimonious parametrization of the correlation structure is desired in order to optimize the efficiency of the estimation procedure we propose a damped exponential correlation structure for modeling multivariate gaussian outcomes the correlation between two observations separated by s units of time is modeled as gamma s theta where gamma is the correlation between elements separated by one s unit and theta is a damping parameter for theta 0 theta 1 and theta infinity the correlation structures of compound symmetry first order autoregressive and first order moving average processes are obtained although the ar 2 dependency structure and the combination of random effects and ar 1 errors are not special cases of the proposed parametric family these structures can be well approximated within the family for short time series maximum likelihood methods for parameter estimation and interpretations of intermediate models 0 less than theta less than 1 are discussed in the context of modeling pulmonary function in an adult population in the netherlands and t cell subsets in homosexual men infected with human immunodeficiency virus type i

a characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere we describe a characterization of convex polyhedra in h 3 in terms of their dihedral angles developed by rivin we also describe some geometric and combinatorial consequences of that theory one of these consequences is a combinatorial characterization of convex polyhedra in e 3 all of whose vertices lie on the unit sphere that resolves a problem posed by jakob steiner in 1832

variational principles for branched circle patterns abstract we are concerned with variational principles for branched circle patterns we use critical point theory to establish existence and uniqueness furthermore we apply the ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric

unconditional almost squareness and applications to spaces of lipschitz functions abstract we introduce an unconditional concept of almost squareness in order to provide a partial negative answer to the problem of existence of almost square dual banach spaces we also take advantage of this notion to provide some criteria of non duality of some subspaces of scalar as well as vector valued lipschitz functions

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

maximum likelihood estimation of intraclass correlation in the analysis of familial data estimating equation approach summary we introduce a generalized model for estimating intraclass correlation in the analysis of familial data this model incorporates the models of donner koval 1980 donner bull 1983 rosner 1984 and miunoz rosner carey 1986 as special cases we then obtain the maximum likelihood estimates of the parameters of the model in particular we obtain a single estimating equation to be solved iteratively for the estimation of the intraclass correlation p and discuss all the special cases asymptotic variance for the estimate of the intraclass correlation is obtained for all the models and some examples are given

advances in the theory and practice of graph drawing the visualization of conceptual structures is a key component of support tools for complex applications in science and engineering foremost among the visual representations used are drawings of graphs and ordered sets in this talk we survey recent advances in the theory and practice of graph drawing specific topics include bounds and tradeoffs for drawing properties three dimensional representations methods for constraint satisfaction and experimental studies

signature of power graphs abstract the signature s g of a graph g is defined as the difference between its positive inertia index and the negative inertia index in 2013 h ma et al 2013 8 conjectured that c 3 g s g c 5 g for an arbitrary simple graph g where c i g denotes the number of cycles in g with length i modulo 4 in 2014 l wang et al 10 proved that c 3 t k s t k c 5 t k for any tree t and for any k 2 in this paper we prove that c 3 g k s g k c 5 g k for any simple graph g and for any k 2 thus extend the main result of 10 to more general cases

relations between the inertia indices of a mixed graph and those of its underlying graph abstract given a graph g the mixed graph d g is obtained from g by orienting some of its edges where g is called the underlying graph of d g let p d g n d g resp p g n g be the positive inertia index and negative inertia index of d g resp g in this paper we first establish the inequalities d g p d g p g d g and d g n d g n g d g where d g is the dimension of cycle space of g furthermore all the corresponding extremal graphs are characterized

a branched andreev thurston theorem for circle packings of the sphere andreev thurston theorem let r be a triangulation of s which is not simplicially equivalent to a tetrahedron and let r 0 n be given where t denotes the set of edges of x suppose that a if eu e2 e3 e r 1 form a closed loop on s and if 2 i e n then eif e2 e3 are the three edges of a two simplex of x b if ex e2 e3 ea e r 1 form a closed loop on s and if 2 i e 2k then ex e2 e2 e4 form the boundary of the union of two adjacent twosimplexes of x

a new approach to the minimum cut problem this paper present a new approach to finding minimum cuts in undirected graphs the fundamental principle is simple the edges in a graph s minimum cut form an extremely small fraction of the graph s edges using this idea we give a randomized strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability the algorithm runs in o n 2 log 3 n time a significant improvement over the previous o mn time bounds based on maximum flows it is simple and intuitive and uses no complex data structures our algorithm can be parallelized to run in rnc with n 2 processors this gives the first proof that the minimum cut problem can be solved in rnc the algorithm does more than find a single minimum cut it finds all of them with minor modifications our algorithm solves two other problems of interest our algorithm finds all cuts with value within a multiplicative factor of of the minimum cut s in expected o n 2 time or in rnc with n 2 processors the problem of finding a minimum multiway cut of graph into r pieces is solved in expected o n 2 r 1 time or in rnc with n 2 r 1 processors the trace of the algorithm s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph this data structure can be efficiently transformed into the more standard cactus representing for minimum cuts

positive and negative inertia index of a graph abstract in this paper the positive and negative inertia index of trees unicyclic graphs and bicyclic graphs are discussed the methods of calculating them are obtained and an inequality about the difference between positive and negative inertia index is proved moreover a conjecture is proposed

analysis of familial data linear model approach summary a linear model approach is proposed for deriving the maximum likelihood estimates of the mother sib interclass correlation and other parameters from familial data when the families have unequal numbers of offspring it leads to an algorithm which involves the maximization of an explicit function of a variable for estimating one parameter and direct substitutions for other parameters the algorithm is believed to be much simpler and more practical than that proposed by rosner 1979 some other advantages of the linearmodel approach are also discussed

regression analysis in the presence of heterogeneous intraclass correlations summary in the statistical analysis of twinship and familial data one often encounters the need for regression methods that control for the possibly different aggregation of the twins and the members of the families according to their type e g monozygotic dizygotic for twinship data we present the maximum likelihood solution for the regression analysis of data in the presence of heterogeneous intraclass correlations this work extends previous results for the case of a homogeneous intraclass correlation an application of the methods for the analysis of twinship data is included

drawing subcubic planar graphs with four slopes and optimal angular resolution abstract a subcubic planar graph is a planar graph whose vertices have degree at most 3 we show that the subcubic planar graphs with at least five vertices have planar slope number at most 4 which is worst case optimal this answers an open question by jelinek et al 10 furthermore we prove that the subcubic planar graphs with at least five vertices have angular resolution 4 which solves an open problem by kant 11 and by formann et al 8

the signature of line graphs and power trees abstract let g be a graph and let a g be the adjacency matrix of g the signature s g of g is the difference between the positive inertia index and the negative inertia index of a g ma et al 2013 10 conjectured that c 3 g s g c 5 g where c 3 g and c 5 g respectively denote the number of cycles in g which have length 4 k 3 and 4 k 5 for some integers k 0 and proved the conjecture holds for trees unicyclic or bicyclic graphs it is known that s g 0 if g is bipartite and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existing results in this paper we show that the conjecture holds for line graphs and power trees

complete metric spaces with property z are length spaces abstract a metric space m is length when every two points have an approximate midpoint while m has property z when for every x y m there exists z such that d x z d z y d x y is arbitrarily smaller than d x z and d z y answering a problem posed by garcia lirola prochazka and rueda zoca we prove that every complete metric space with property z is length as a consequence we get that for the lipschitz free space of a metric space a number of geometric properties are all equivalent like the daugavet property the diameter 2 property the absence of strongly exposed points or the absence of preserved extreme points

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

radiocoloring in planar graphs complexity and approximations the frequency assignment problem fap in radio networks is the problem of assigning frequencies to transmitters by exploiting frequency reuse while keeping signal interference to acceptable levels the fap is usually modelled by variations of the graph coloring problem a radiocoloring rc of a graph g v e is an assignment function v n such that u v 2 when u v are neighbors in g and u v 1 when the distance of u v in g is two the number of discrete frequencies and the range of frequencies used are called order and span respectively the optimization versions of the radiocoloring problem rcp are to minimize the span or the order in this paper we prove that the radiocoloring problem for general graphs is hard to approximate unless np zpp within a factor of n1 2 e for any e 0 where n is the number of vertices of the graph however when restricted to some special cases of graphs the problem becomes easier we prove that the min span rcp is np complete for planar graphs next we provide an o n time algorithm v n which obtains a radiocoloring of a planar graph g that approximates the minimum order within a ratio which tends to 2 where the maximum degree of g finally we provide a fully polynomial randomized approximation scheme fpras for the number of valid radiocolorings of a planar graph g with colors in the case where 4 50

lenses and asymptotic midpoint uniform convexity abstract we introduce a new isometric geometric property namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties

on embedding a graph in the grid with the minimum number of bends given a planar graph g together with a planar representation p a region preserving grid embedding of g is a planar embedding of g in the rectilinear grid that has planar representation isomorphic to p in this paper an algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges this algorithm makes use of network flow techniques and runs in time o n 2 log n where n is the number of vertices of the graph constrained versions of the problem are also considered and most results are extended to k gonal graphs i e graphs whose edges are sequences of segments with slope multiple of 180 k degrees applications of the above results can be found in several areas vlsi circuit layout architectural design communication by light or microwave transportation problems and automatic layout of graphlike diagrams

drawing graphs in the plane with high resolution this paper presents the problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized the resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph and the resolution of a graph to be the maximum resolution of any layout of the graph the resolution r of a graph is characterized in terms of the maximum node degree d of the graph by proving that omega frac 1 d 2 leqslant r leqslant frac 2 pi d for any graph moreover it is proved that r theta frac 1 d for many graphs including planar graphs complete graphs hypercubes multidimensional meshes and tori and other special networks it is also shown that the problem of deciding if r frac 2 pi d for a graph is np hard for d 4 and by using a counting argument that r o frac log d d 2 for many graphs

a characterisation of the daugavet property in spaces of lipschitz functions abstract we study the daugavet property in the space of lipschitz functions lip 0 m on a complete metric space m namely we show that lip 0 m has the daugavet property if and only if m is a length metric space this condition also characterises the daugavet property in the lipschitz free space f m moreover when m is compact we show that either f m has the daugavet property or its unit ball has a strongly exposed point if m is an infinite compact subset of a strictly convex banach space then the daugavet property of lip 0 m is equivalent to the convexity of m

multivariate methods in ophthalmology with application to other paired data situations methods are presented for performing multiple regression analyses and multiple logistic regression analyses on ophthalmologic data with normally and binomially distributed outcome variables while accounting for the intraclass correlation between eyes these methods are extended to more general nested data structures where a variable number of subunits are available for each primary unit of analysis as in familial data these methods can also be applied to other types of paired data as in matched studies with a variable matching ratio where one has a continuous outcome variable and wishes to control for other confounding variables while maintaining the matching examples are given of these methods with a group of over 400 patients with retinitis pigmentosa in which spherical refractive error and visual acuity are related to genetic type after the effects of age sex and the presence of cataract have been controlled

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

points of differentiability of the norm in lipschitz free spaces abstract we consider convex series of molecules in lipschitz free spaces i e elements of the form n n x n y n d x n y n such that n n we characterise these elements in terms of geometric conditions on the points x n y n of the underlying metric space and determine when they are points of g teaux differentiability of the norm in particular we show that g teaux and frechet differentiability are equivalent for finitely supported elements of lipschitz free spaces over uniformly discrete and bounded metric spaces and that their tensor products with g teaux resp frechet differentiable elements of a banach space are g teaux resp frechet differentiable in the corresponding projective tensor product

supports in lipschitz free spaces and applications to extremal structure abstract we show that the class of lipschitz free spaces over closed subsets of any complete metric space m is closed under arbitrary intersections improving upon the previously known finite diameter case this allows us to formulate a general and natural definition of supports for elements in a lipschitz free space f m we then use this concept to study the extremal structure of f m we prove in particular that x y d x y is an exposed point of the unit ball of f m whenever the metric segment x y is trivial and that any extreme point which can be expressed as a finitely supported perturbation of a positive element must be finitely supported itself we also characterize the extreme points of the positive unit ball they are precisely the normalized evaluation functionals on points of m

the bishop phelps bollobas property for lipschitz maps abstract in this paper we introduce and study a lipschitz version of the bishop phelps bollobas property lip bpb property this property deals with the possibility of making a uniformly simultaneous approximation of a lipschitz map f and a pair of points at which f almost attains its norm by a lipschitz map g and a pair of points such that g strongly attains its norm at the new pair of points we first show that if m is a finite pointed metric space and y is a finite dimensional banach space then the pair m y has the lip bpb property and that both finiteness assumptions are needed next we show that if m is a uniformly gromov concave pointed metric space i e the molecules of m form a set of uniformly strongly exposed points then m y has the lip bpb property for every banach space y we further prove that this is the case for finite concave metric spaces ultrametric spaces and holder metric spaces the extension of the lip bpb property from m r to some banach spaces y and some results for compact lipschitz maps are also discussed

octahedrality in lipschitz free banach spaces the aim of this note is to study octahedrality in vector valued lipschitz free banach spaces on a metric space under topological hypotheses on it by analysing the weak star strong diameter 2 property in lipschitz function spaces also we show an example that proves that our results are optimal and that octahedrality in vector valued lipschitz free banach spaces actually relies on the underlying metric space as well as on the banach one

on line planarity testing the on line planarity testing problem consists of performing the following operations on a planar graph g i testing if a new edge can be added to g so that the resulting graph is itself planar ii adding vertices and edges such that planarity is preserved an efficient technique for on line planarity testing of a graph is presented that uses o n space and supports tests and insertions of vertices and edges in o log n time where n is the current number of vertices of g the bounds for tests and vertex insertions are worst case and the bound for edge insertions is amortized we also present other applications of this technique to dynamic algorithms for planar graphs

algorithms for plane representations of acyclic digraphs abstract acyclic digraphs are widely used for representing hierarchical structures examples include pert networks subroutine call graphs family trees organization charts hasse diagrams and isa hierarchies in knowledge representation diagrams we investigate the problem of representing acyclic digraphs in the plane in such a way that all edges flow in the same direction e g from the left to the right or from the bottom to the top three plane representations are considered straight drawings visibility representations and grid drawings we provide efficient algorithms that construct these representations with all edges flowing in the same direction the time complexity is o n for visibility representations and grid drawings and o n log n for straight drawings where n is the number of vertices of the digraph for covering digraphs of lattices the complexity of constructing straight drawings is o n we also show that the planar digraphs that admit any one of these representations are exactly the subgraphs of planar st graphs

octahedral norms in duals and biduals of lipschitz free spaces abstract we continue with the study of octahedral norms in the context of spaces of lipschitz functions and in their duals first we prove that the norm of f m is octahedral as soon as m is unbounded or is not uniformly discrete further we prove that a concrete sequence of uniformly discrete and bounded metric spaces k m satisfies that the norm of f k m is octahedral for every m finally we prove that if x is an arbitrary banach space and the norm of lip 0 m is octahedral then the norm of lip 0 m x is octahedral these results solve several open problems from the literature

banach spaces with the daugavet property a banach space x is said to have the daugavet property if every operator t x x of rank 1 satisfies 11 id tl 1 fltil we show that then every weakly compact operator satisfies this equation as well and that x contains a copy of t1 however x need not contain a copy of l1 we also study pairs of spaces x c y and operators t x y satisfying i i j t i i 1 4 1f t i i where j x y is the natural embedding this leads to the result that a banach space with the daugavet property does not embed into a space with an unconditional basis in another direction we investigate spaces where the set of operators with f id tll 1 il t l i is as small as possible and give characterisations in terms of a smoothness condition