minimal cost brother trees we investigate three cost measures for the recently introduced brother search trees in particular we characterize node visit optimal comparison cost optimal and space cost optimal 1 2 brother trees and present linear time algorithms to construct optimal 1 2 brother trees for each cost measure furthermore we also consider briefly these cost measures for brother leaf search trees

higher dimensional algebra i braided monoidal 2 categories abstract we begin with a brief sketch of what is known and conjectured concerning braided monoidal 2 categories and their relevance to 4d tqfts and 2 tangles then we give concise definitions of semistrict monoidal 2 categories and braided monoidal 2 categories and show how these may be unpacked to give long explicit definitions similar to but not quite the same as those given by previously kapranov and voevodsky finally we describe how to construct a semistrict braided monoidal 2 category z c as the center of a semistrict monoidal category c in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category as a corollary this yields a strictification theorem for braided monoidal 2 categories

counting lattice points of rational polyhedra abstract the generating function f p p zn x for a rational polytope p carries all essential information of p in this paper we show that for any positive integer n the generating function f p n of np nx x p can be written as f p n a p n x n where a is the set of all vertices of p and each p n is a certain periodic function of n the ehrhart reciprocity law follows automatically from the above formula we also present a formula for the coefficients of ehrhart polynomials in terms of elementary symmetric functions

cubic hypersurfaces admitting an embedding with gauss map of rank 0 abstract we give a characterization of fermat cubic hypersurfaces of dimension greater than 2 in characteristic 2 in terms of the property called gmrz that a projective variety admits an embedding whose gauss map is of rank 0 in contrast to the higher dimensional case for cubic surfaces the above characterization is no longer true moreover we prove that the process of blowing up at points preserves the property gmrz and that every smooth rational surface in fact satisfies gmrz in the characteristic 2 case

lattice points in simple polytopes in terms of fp h q x dx where the polytope p h is obtained from p by independent parallel motions of all facets this extends to simple lattice polytopes the eulermaclaurin summation formula of khovanskii and pukhlikov 8 valid for lattice polytopes such that the primitive vectors on edges through each vertex of p form a basis of the lattice as a corollary we recover results of pommersheim 9 and kantor khovanskii 6 on the coefficients of the ehrhart polynomial of p our proof is elementary in a subsequent article we will show how to adapt it to compute the equivariant todd class of any complete toric variety with quotient singularities the euler maclaurin summation formula for simple lattice polytopes has been obtained independently by ginzburg guillemin karshon 4 they used the dictionary between convex polytopes and projective toric varieties with an ample divisor class in combination with the riemann roch kawasaki formula 1 7 for complex manifolds with quotient singularities a counting formula for lattice points in lattice simplices has been announced by cappell and shaneson 2 as a consequence of their computation of the todd class of toric varieties with quotient singularities

values of zeta functions at negative integers dedekind sums and toric geometry in the present paper we study relations among special values of zeta functions of real quadratic fields properties of generalized dedekind sums and todd classes of toric varieties the main theme of the paper is the use of toric geometry to explain in a conceptual way properties of the values of zeta functions and dedekind sums as well as to provide explicit computations both toric varieties and zeta functions associate numerical invariants to cones in lattices with different motivations and applications though we will focus on the case of two dimensional cones in the present paper we introduce notation and definitions that are valid for cones of arbitrary dimension the reason for this added generality is clarity as well as preparation for the results of a subsequent publication

loop spaces and coherence for monoidal and braided monoidal bicategories abstract we prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories we show how coherence for these structures can be interpreted topologically using up to homotopy operad actions and the algebraic classification of surface braids

computing the ehrhart quasi polynomial of a rational simplex we present a polynomial time algorithm to compute any fixed number of the highest coefficients of the ehrhart quasi polynomial of a rational simplex previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension the algorithm is based on the formula relating the kth coefficient of the ehrhart quasi polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to k dimensional faces of the polytope we discuss possible extensions and open questions

monoidal bicategories and hopf algebroids why are groupoids such special categories the obvious answer is because all arrows have inverses yet this is precisely what is needed mathematically to model symmetry in nature the relation between the groupoid and the physical object is expressed by an action the presence of inverses means that actions of a groupoid g behave much better than actions of an arbitrary category the totality of actions of g on vector spaces forms a category mod g of modules the feature of mod g which epitomises the fact that g is a groupoid is that the internal hom in mod g is calculated in a particularly simple way more precisely the functor out of mod g which forgets the actions preserves not only the monoidal structure but also the closed structure with this as a guiding principle we develop a general concept of autonomous pseudomonoid which includes ordinary hopf algebras indeed hopf algebroids and autonomous compact rigid monoidal categories this is intended to elucidate the interaction between hopf algebras and autonomous monoidal categories in tannaka duality as appearing in js2 d2 for example given a topological monoid m it is explained in js2 section 8 why the monoidal category of finite dimensional representations of m is equivalent to the monoidal category of finite dimensional comodules over the bialgebra r m of representative functions on m this provides evidence that when regarding a hopf algebra as a quantum group it is the finitedimensional comodules rather than modules which should be regarded as the representations of the group in dealing with comodules we are using the coalgebra structure of the hopf algebra h a hopf algebroid is an additive category that is algebra with several objects that is algebroid with a comonoidal structure modules over a hopf algebroid make sense but comodules are not appropriate what we need is a notion where the article no ai971649

generalized centers of braided and sylleptic monoidal 2 categories abstract recent developments in higher dimensional algebra due to kapranov and voevodsky day and street and baez and neuchl include definitions of braided sylleptic and symmetric monoidal 2 categories and a center construction for monoidal 2 categories which gives a braided monoidal 2 category i give generalized center constructions for braided and sylleptic monoidal 2 categories which give sylleptic and symmetric monoidal 2 categories respectively and i correct some errors in the original center construction for monoidal 2 categories

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

a tight upper bound for the path length of avl trees we prove that the internal path length of an avl tree of size n is bounded from above by 1 4404n log 2 n log 2 log 2 n o n and show that this bound is achieved by an infinite family of avl trees but avl trees of maximal height do not have maximal path length these results carry over to the comparison cost of brother trees

on the path length of binary trees it is shown that the external path length of a binary tree is closely related to the ratios of means of certain integers and establish the upper bound external path length n log 2 n d log 2 d 0 6623 where n denotes the number of external nodes in the tree and d is the difference in length between a longest and shortest path then it is proved that this bound is tight up to an o n term if d n if d n we contstruct binary trees whose external path length is at least as large as n log 2 n f n d d log 2 d 4 where f n d 1 1 2 d n

the euler maclaurin formula for simple integral polytopes we give a euler maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope our proof uses elementary methods

the ehrhart polynomial of a lattice polytope the problem of counting the number of lattice points inside a convex lattice polytope in r a polytope whose vertices have integer coordinates has been studied from a variety of perspectives here we use the poisson summation formula and related techniques in fourier analysis to obtain a formula for the number of lattice points inside all integral dilates of a lattice polytope we show that an associated generating function has an explicit representation in terms of sums of cotangents which are higher dimensional generalizations of dedekind sums let en denote the n dimensional integer lattice in r and let p be an n dimensional lattice polytope in r which is a compact simplicial complex of pure dimension n whose vertices lie on the lattice consider the function of an integer valued variable t that describes the number of lattice points that lie inside the dilated polytope tp

classifying spaces for braided monoidal categories and lax diagrams of bicategories abstract this work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces bicategories in particular monoidal categories have well understood simple geometric realizations and we here deal with homotopy types represented by lax diagrams of bicategories that is lax functors to the tricategory of bicategories in this paper it is proven that when a certain bicategorical grothendieck construction is performed on a lax diagram of bicategories then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram this result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type up to group completion as the underlying category of any given non necessarily strict braided monoidal category specifically it is proven that these double delooping spaces for categories enriched with a braided monoidal structure can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories which we refer to as their street s geometric nerves

balance in avl trees and space cost of brother trees abstract we characterize avl trees that have for their heights and weights the maximum numbers of nodes whose subtrees differ in height by one imbalanced nodes we obtain the result from a characterization of brother trees with the maximum space costs for their heights and weights the proof is based on a novel tree transformation that is of interest in its own right

k theory for 2 categories abstract we establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra for this we develop the theory of objects in 2 categories in the course of the proof we establish strictification results of independent interest for symmetric monoidal bicategories and for diagrams of 2 categories

infinite loop spaces and coherence for symmetric monoidal bicategories abstract this paper proves three different coherence theorems for symmetric monoidal bicategories first we show that in a free symmetric monoidal bicategory every diagram of 2 cells commutes second we show that this implies that the free symmetric monoidal bicategory on one object is equivalent as a symmetric monoidal bicategory to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups third we show that every symmetric monoidal bicategory is equivalent to a strict one we give two topological applications of these coherence results first we show that the classifying space of a symmetric monoidal bicategory can be equipped with an e structure second we show that the fundamental 2 groupoid of an e n space n 4 has a symmetric monoidal structure these calculations also show that the fundamental 2 groupoid of an e 3 space has a sylleptic monoidal structure