on structure sensitivity of degree based topological indices one of the general requirements for any topological index ti is that similar molecules have near lying ti values which is referred to as smoothness curiously however smoothness of topological indices was until now never examined and never quantified we now propose a pertinent mathematical model for this property and undertake a comparative study of the structure sensitivity of 12 degree based topological indices by using trees

classifying spaces of kac moody groups we study the structure of classifying spaces of kac moody groups from a homotopy theoretic point of view they behave in many respects as in the compact lie group case the mod p cohomology algebra is noetherian and lannes t functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p subgroups also spaces of maps from classifying spaces of finite p groups to classifying spaces of kac moody groups are described in terms of classifying spaces of centralizers while the classifying space of a kac moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p subgroups up to p completion we show that these properties are common to a larger class of groups also including parabolic subgroups of kac moody groups and centralizers of finite p subgroups

efficient computation of trees with minimal atom bond connectivity index the atom bond connectivity abc index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry for a graph g the abc index is defined as u v e g d u d v 2 d u d v where d u is the degree of vertex u in g and e g is the set of edges of g despite many attempts in the last few years it is still an open problem to characterize trees with minimal abc index in this paper we present an efficient approach of computing trees with minimal abc index by considering the degree sequences of trees and some known properties of the trees with minimal abc index the obtained results disprove some existing conjectures and suggest new ones to be set

on topological properties of the line graphs of subdivision graphs of certain nanostructures in the study of qsar qspr topological indices such as shultz index generalized randic index zagreb index general sum connectivity index atom bond connectivity abc index and geometric arithmetic ga index are exploited to estimate the bioactivity of chemical compounds a topological index attaches a chemical structure with a numeric number there are numerous applications of graph theory in this field of research in this paper we computed generalized randic general zagreb general sum connectivity abc ga abc4 and ga5 indices of the line graphs of 2d lattice nanotube and nanotorus of tuc4c8p q by using the concept of subdivision

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

equivalences between fusion systems of finite groups of lie type when g is a finite group and p is a prime the fusion system fp g is the category whose objects are the p subgroups of g and whose morphisms are the homomorphisms between subgroups induced by conjugation in g if g is another finite group then fp g and fp g are isotypically equivalent if there is an equivalence of categories between them which commutes up to natural isomorphism of functors with the forgetful functors from fp to the category of groups alternatively fp g and fp g are isotypically equivalent if there is an isomorphism between sylow p subgroups of g and of g which is fusion preserving in the sense of definition 1 2 below the goal of this paper is to use methods from homotopy theory to prove that certain pairs of fusion systems of finite groups of lie type are isotypically equivalent our main result is the following theorem

on the uniqueness of possibilistic measure of uncertainty and information abstract it is demonstrated through a series of theorems that the u uncertainty introduced by higashi and klir in 1982 is the only possibilistic measure of uncertainty and information that satisfies possibilistic counterparts of axioms of the well established shannon and hartley measures of uncertainty and information two complementary forms of the possibilistic counterparts of the probabilistic branching or grouping axiom which is usually used in proofs of the uniqueness of the shannon measure are introduced in this paper for the first time a one to one correspondence between possibility distributions and basic probabilistic assignments introduced by shafer in his mathematical theory of evidence is instrumental in most proofs in this paper the uniqueness proof is based on possibilistic formulations of axioms of symmetry expansibility additivity branching monotonicity and normalization

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

finite loop spaces are manifolds one of the motivating questions for surgery theory was whether every finite h space is homotopy equivalent to a lie group this question was answered in the negative by hilton and roitberg s discovery of some counterexamples 18 however the problem remained whether every finite h space is homotopy equivalent to a closed smooth manifold this question is still open but in case the h space admits a classifying space we have the following theorem

stratification and duality for homotopical groups abstract we generalize quillen s f isomorphism theorem quillen s stratification theorem the stable transfer and the finite generation of cohomology rings from finite groups to homotopical groups as a consequence we show that the category of module spectra over c b g f p is stratified and costratified for a large class of p local compact groups g including compact lie groups connected p compact groups and p local finite groups thereby giving a support theoretic classification of all localizing and colocalizing subcategories of this category moreover we prove that p compact groups admit a homotopical form of gorenstein duality

normalizers of maximal tori normalizers and p normalizers of maximal tori in p compact groups can be characterized by the euler characteristic of the associated homogeneous spaces applied to centralizers of elementary abelian p groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus i e that normalizer commutes with centralizer

vertex degree based topological indices of hexagonal systems with equal number of edges in this paper we find extremal values of vertex degree based topological indices over m the set of hexagonal systems with m edges the main idea consists in constructing hexagonal systems with maximal number of inlets in m which have simultaneously minimal number of hexagons also we show that the convex spiral sh has extremal ti value over 3h 12h3

rings of generalized and stable invariants of pseudoreflections and pseudoreflection groups let g gl n f be a representation of a finite groupgover the field f and f v the space of polynomial functions onv fn we associate togan idealj g f v called the ideal of stable invariants ofg or more precisely if s gl n f is a set of pseudoreflections we associate to s an ideali s f v called the ideal of generalized invariants of s whengis a pseudoreflection group we investigatei s for various choices of s gand the relation betweenj g andi s to a representation g gl n f of a finite group respectively to a set s gl n f of pseudoreflections we also associate a ring formula respectively formula we show that formula is always a polynomial algebra over f and whenever g is generated by semisimple pseudoreflections s that formula

graphs with fixed number of pendent vertices and minimal zeroth order general randic index we investigate the graph with minimal zeroth order general randic index in terms of its order n pendent number n1 and cyclomatic number 0 extremal graphs were completely characterized for cases of 0 1 2 which can be directly extended for graphs with cyclomatic number 3

sharp lower bounds on the narumi katayama index of graph operations abstract the narumi katayama index nk g of a simple graph g is equal to the product of the degrees of the vertices of g in this paper we present sharp lower bounds on the narumi katayama index of several graph operations such as union join suspension rooted product cluster corona product direct product cartesian product strong product generalized hierarchical product composition disjunctive product and symmetric difference in terms of the narumi katayama indices and the orders of their components using these results the narumi katayama index of several classes of graphs will be computed

quillen grassmannians as non modular homotopy fixed points the aim of this paper is to compute the homotopy fixed points of the homotopy action described by unstable adams operations on the classifying spaces of unitary groups the same technique can be applied to compute homotopy fixed points of the action of certain automorphisms on p compact groups called generalized grassmannians we use the description of quillen grassmannians to describe the set of homotopy representations of elementary abelian p groups into them

relations between distance based and degree based topological indices let w sz pi and wp be respectively the wiener szeged pi and wiener polarity indices of a molecular graph g let m1 and m2 be the first and second zagreb indices of g we obtain relations between these classical distance and degree based topological indices

vector bundles over davis januszkiewicz spaces with prescribed characteristic classes for any n 1 dimensional simplicial complex we construct a particular n dimensional complex vector bundle over the associated davis januszkiewicz space whose chern classes are given by the elementary symmetric polynomials in the generators of the stanley reisner algebra we show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes this allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa similar questions are also discussed for 2n dimensional real vector bundles with particular prescribed characteristic pontrjagin and euler classes we also analyze which of these bundles admit a complex structure it turns out that all these bundles are closely related to the tangent bundles of quasi toric manifolds and moment angle complexes 2012 american mathematical society

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

homotopy nilpotency in p regular loop spaces we consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p completion of a product of finite numbers of spheres we determine the homotopy nilpotency of those loop spaces as an answer to this problem

degree based topological indices optimal trees with given number of pendents abstract a dynamic programming method is elaborated enabling the characterization of trees with a given number of pendent vertices for which a vertex degree based invariant topological index achieves its extremal value the method is applied to the chemically interesting and earlier much studied such invariants the first and second zagreb index and the atom bond connectivity index

on zagreb indices zagreb polynomials of some nanostar dendrimers dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core ashrafi and mirzargar 2008 3 chen et al 2014 7 klajnert and bryszewska 2001 26 yamamoto et al 2002 36 these are key molecules in nanotechnology and can be put to good use topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure activity property toxicity relationships these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds in this paper we determine hyper zagreb index first multiple zagreb index second multiple zagreb index and zagreb polynomials for some nanostar dendrimers

on structural properties of abc minimal chemical trees abstract the atom bond connectivity abc index of a connected graph g is defined as a b c g u v e g d u d v 2 d u d v where d w is the degree of vertex w in g and e g is the set of edges of g given greedy trees perturbed by different degree sequences we investigate the behaviors on their abc index in this paper moreover we also give the minimum abc index and its structural properties of the chemical trees with n vertices and p pendant vertices for n 3 p 2

extremal values of vertex degree based topological indices over hexagonal systems with fixed number of vertices let g be a graph with n vertices a vertex degree based topological index is defined from a set of real numbers j i j email protected 1

on abc eigenvalues and abc energy abstract let g be a graph with n vertices and let d i be the degree of its i th vertex the abc matrix of g is the square matrix of order n whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise this matrix related closely to the atom bond connectivity abbreviated as abc index was recently introduced by estrada as a matrix representation of the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph which in the context of molecular graphs can be related to the polarizing capacity of the bond considered the abc eigenvalues of g are the eigenvalues of its abc matrix and the abc energy of g is the sum of the absolute values of its abc eigenvalues in this paper some new results for the abc eigenvalues and the abc energy of a graph are presented

on structural properties of trees with minimal atom bond connectivity index iv solving a conjecture about the pendent paths of length three abstract the atom bond connectivity abc index is one of the most investigated degree based molecular structure descriptors with a variety of chemical applications it is known that among all connected graphs the trees minimize the abc index however a full characterization of trees with a minimal abc index is still an open problem by now one of the proved properties is that a tree with a minimal abc index may have at most one pendent path of length 3 with the conjecture that it cannot be a case if the order of a tree is larger than 1178 here we provide an affirmative answer of a strengthened version of that conjecture showing that a tree with minimal abc index cannot contain a pendent path of length 3 if its order is larger than 415

probabilistic inequalities for evaluating structural network measures proving interrelations between structural graph measures analytically has been intricate generally relations between structural graph measures describe the interplay between measures which turned out to be useful for better understanding the properties of such quantities the results which have been achieved so far are restricted to simple measures or special graph classes such as trees in this paper we introduce a probabilistic approach for establishing inequalities between quantitative network measures on random networks those inequalities between different graph measures lead to a deeper understanding of the mathematical apparatus and in particular to a discussion of quality aspects of structural graph measures which is a major contribution of this paper

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

zagreb indices of transformation graphs and total transformation graphs the first zagreb index of a graph g is the sum of squares of the degrees of the vertices of g expressions for the first zagreb index and first zagreb coindex are obtained for total semitotal line or middle and semitotal point graphs as well as of eight distinct types of total transformation graphs g xyz

computer search for trees with minimal abc index the abc index is a degree based molecular structure descriptor that found chemical applications finding the connected graph s of a given order whose abc index is minimal is a hitherto unsolved problem but it is known that these must be trees in this paper by combining mathematical arguments and computer based modeling we establish the basic structural features of the minimum abc trees

reconstruction families of possibilistic structure systems abstract a method is presented for characterizing the family of overall systems reconstructable from a given possibilistic structure system the technique is elaborated in terms of fuzzy relation equations it is demonstrated that the reconstruction family of a given structure system is equivalent to the set of solutions of a special type of fuzzy relation equation the solution set is partially ordered and contains both minimal solutions and a unique maximum solution when these elements are identified all members of the reconstruction family are determined another characteristic of the reconstruction family is its reconstruction uncertainty a measure of which is also developed in this paper this measure is used to define an identifiability quotient that expresses the degree of confidence with which we may identify a single overall system given a particular structure system

on the edge version atom bond connectivity and geometric arithmetic indices of certain graph operations let dl g e be the degree of an edge e in line graph l g of a graph g the edge versions of atom bond connectivity abce and geometric arithmetic gae indices of g are defined as efe l g dl g e dl g f 2dl g e dl g f and gae g efe l g 2dl g e dl g f dl g e dl g f in this paper we study the abce and gae indices for joint graphs and certain graph operations

on structural properties of trees with minimal atom bond connectivity index iii the atom bond connectivity abc index is a degree based graph topological index that found chemical applications including those of predicting the stability of alkanes and the strain energy of cycloalkanes several structural properties of the trees with minimal abc index were proved recently however the complete characterization of the minimal abc trees is still an open problem it is known that minimal abc trees can have at most one pendent path of length 3 it is also known that the minimal abc trees that have a pendent path of length 3 do not contain so called bk branches with k 4 and do not contain more than two b2 branches here we improve the latter result by showing that minimal abc trees of order larger than 168 and with a pendent path of length 3 do not contain b2 branches moreover we show that trees with minimal abc index with a pendent path of length 3 do not contain b1 branches

on the maximum abc index of graphs without pendent vertices abstract let g be a simple graph the atom bond connectivity index abc of g is defined as a b c g u v e g d u d v 2 d u d v where d v denotes the degree of vertex v of g we characterize the graphs with n vertices minimum vertex degree 2 and m edges for m 2 n 4 and m 2 n 3 that have maximum abc index

h spaces analogous to e 8 mod 3 let p be an odd prime let x0 be a finite p local simply connected homotopy associative h space suppose h x0 p contains the subalgebra open image in new window satisfying open image in new window the only known examples occur for p 3 and involve the lie group e8 in this note we prove that if x0 exists then p must be 3

loop structures on homotopy fibers of self maps of spheres let s 2 n 1 k denote the fiber of the degree k map on the sphere s 2 n 1 if k p r where p is an odd prime and n divides p 1 then s 2 n 1 k is known to be a loop space it is also known that s 3 2 r is a loop space for r 3 in this paper we study the possible loop structures on this family of spaces for all primes p in particular we show that s 3 4 is not a loop space our main result is that whenever s 2 n 1 p r is a loop space the loop structure is unique up to homotopy

remarks on the graovac ghorbani index of bipartite graphs the atom bond connectivity abc index is a well known degree based molecular structure descriptor with a variety of chemical applications in 2010 graovac and ghorbani introduced a distance based analog of this index the graovac ghorbani gg index which yielded promising results when compared to analogous descriptors in this paper we investigate the structure of graphs that maximize and minimize the gg index specifically we show that amongst all bipartite graphs the minimum gg index is attained by a complete bipartite graph while the maximum gg index is attained by a path or a cycle like graph the structure of the resulting graph depends on the number of vertices through the course of the research we also derive an asymptotic estimate of the gg index of paths in order to obtain our results we introduce a normalized version of the gg index and call it the normalized graovac ghorbani ngg index finally we discuss some related open questions as a potential extension of our work

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

a finite loop space not rationally equivalent to a compact lie group we construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact lie group hence providing a counterexample to a classical conjecture aided by machine calculation we verify that our counterexample is minimal i e that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact lie group extending the classical known bound of 5

where do we stand on measures of uncertainty ambiguity fuzziness and the like abstract it is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty each of these types requires a measure by which the degree of uncertainty of that type can be determined two main categories of uncertainty are connected with the terms vagueness or fuzziness and ambiguity in general vagueness is associated with the difficulty of making sharp or precise distinctions in the world ambiguity on the other hand is associated with one to many relations i e situations with two or more alternatives that are left unspecified while the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness the concept of a fuzzy measure is a general framework for dealing with ambiguity several classes of measures of vagueness usually referred to as measures of fuzziness have been proposed in the literature each class is based on some underlying conception of the degree of fuzziness a general set of requirements for measures of fuzziness is formulated followed by an overview of the measures proposed in the literature measures of ambiguity are discussed within the framework of plausibility and belief measures although it does not cover all fuzzy measures this framework is sufficiently broad for most practical purposes and represents a generalization of both probability theory and possibility theory it is argued that three complementary measures of ambiguity should be employed one of them is obtained by generalizing the hartley measure of uncertainty it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures the other two are obtained by generalizing the well known shannon measure of uncertainty they measure the degree of dissonance and the degree of confusion in evidence respectively basic mathematical properties of these measures are overviewed it is also argued that each of the four types of uncertainty measures which are fundamentally different from each other can be used for measuring structural syntactic information in the same sense as the hartley and shannon measures have been used in this respect as such these measures are potentially powerful tools for dealing with systems problems such as systems modelling analysis simplification or design

on the cohomology of highly connected covers of finite hopf spaces relying on the computation of the andre quillen homology groups for unstable hopf algebras we prove that if the mod p cohomology of both the fiber and the base in an h fibration is finitely generated as algebra over the steenrod algebra then so is the mod p cohomology of the total space in particular the mod p cohomology of the n connected cover of a finite h space is always finitely generated as algebra over the steenrod algebra

on extremality of abc spectral radius of a tree abstract the abc matrix of a graph g recently introduced by estrada is the square matrix of order g whose i j entry is equal to d i d j 2 d i d j if the i th vertex and the j th vertex of g are adjacent and 0 otherwise where d i is the degree of the i th vertex of g the abc spectral radius of g is the largest eigenvalue of the abc matrix of g which is denoted by g in this paper we prove that for any tree t of order n 3 2 cos n 1 t n 2 with equality in the left resp right inequality if and only if t is isomorphic to the path p n resp the star k 1 n 1

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

on the mod p cohomology of bpu p we study the mod p cohomology of the classifying space of the projective unitary group pu p we first prove that conjectures due to j f adams and kono and yagita 1993 about the structure of the mod p cohomology of the classifying space of connected compact lie groups hold in the case of pu p finally we prove that the classifying space of the projective unitary group pu p is determined by its mod p cohomology as an unstable algebra over the steenrod algebra for p 3 completing previous work by dwyer miller and wilkerson 1992 and broto and viruel 1998 for the cases p 2 3

generalizations of szőkefalvi nagy and chebyshev inequalities with applications in spectral graph theory abstract two weighted inequalities for real non negative sequences are proven the first one represents a generalization of the sz kefalvi nagy inequality for the variance and the second a generalization of the discrete chebyshev inequality for two real sequences then the obtained inequalities are used to determine lower bounds for some degree based topological indices of graphs

reconstruction of possibilistic behavior systems abstract reconstructability analysis is viewed as a process of investigating the possibilities of reconstructing desirable properties of overall systems from the knowledge of the corresponding properties of their various subsystems the reconstructability analysis consists of procedures for generating meaningful reconstruction hypotheses procedures for the evaluation of the reconstruction hypotheses and procedures for making various decisions regarding the acceptance of evaluated reconstruction hypotheses generation of additional reconstruction hypotheses termination of the analysis and the like the paper discusses the evaluation of reconstruction hypotheses when the systems under consideration are possibilistic behavior systems it is shown that a principle of maximum ambiguity similar to the principle of maximum entropy for probabilistic systems can be used for possibilistic systems it is also shown that the unbiased maximum ambiguity reconstruction can be determined by a simple join procedure in a similar fashion as for probabilistic systems the join procedure for possibilistic systems turns out to be computationally simpler than the one for probabilistic systems the paper also describes a general procedure for determining the reconstruction family

homology decompositions for p compact groups we construct a homotopy theoretic setup for homology decompositions of classifying spaces of p compact groups this setup is then used to obtain a subgroup decomposition for p compact groups which generalizes the subgroup decomposition with respect to p stubborn subgroups for a compact lie group constructed by jackowski mcclure and oliver

on the p compact groups corresponding to the p adic reflection groups g q r n there exists an infinite family of p compact groups whose weyl groups correspond to the finite p adic pseudoreflection groups g q r n of family 2a in the clark ewing list in this paper we study these p compact groups in particular we construct an analog of the classical whitney sum map a family of monomorphisms and a spherical fibration which produces an analog of the classical j homomorphism finally we also describe a faithful complexification homomorphism from these p compact groups to the p completion of unitary compact lie groups

gkm theory for p compact groups abstract this work studies the flag varieties of p compact groups principally through torus equivariant cohomology extending methods and tools of classical schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups in particular we give for the infinite family of p compact flag varieties corresponding to the complex reflection groups g r 1 n a generalized gkm characterization following goresky kottwitz macpherson 8 of the torus equivariant cohomology building an explicit additive basis and showing its relationship with the polynomial or borel presentation via the localization map

quantitative graph theory in this paper we describe some highlights of the new branch quantitative graph theory and explain its significant different features compared to classical graph theory the main goal of quantitative graph theory is the structural quantification of information contained in complex networks by employing a measurement approach based on numerical invariants and comparisons furthermore the methods as well as the networks do not need to be deterministic but can be statistic as such this complements the field of classical graph theory which is descriptive and deterministic in nature we provide examples of how quantitative graph theory can be used for novel applications in the context of the overarching concept network science

connected finite loop spaces with maximal tori finite loop spaces are a generalization of compact lie groups however they do not enjoy all of the nice properties of compact lie groups for example having a maximal torus is a quite distinguished property actually an old conjecture due to wilkerson says said that every connected finite loop space with a maximal torus is equivalent to a compact connected lie group we give some more evidence for this conjecture by showing that the associated action of the weyl group on the maximal torus always represents the weyl group as a crystallographic group we also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected lie group if it has the right normalizer of the maximal torus actually in the cases under consideration the information about the weyl group is sufficient to give the answer all this is done by first studying the analoguous local problems