beta integers as natural counting systems for quasicrystals recently discrete sets of numbers the integers have been proposed as numbering tools in quasicrystalline studies indeed there exists a unique numeration system based on the irrational in which the integers are all real numbers with no fractional part these integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to golden mean to or to i e when is one of the self similarity ratios observed in quasicrystalline structures as a matter of fact integers are natural candidates for coordinating quasicrystalline nodes and also the bragg peaks beyond a given intensity in corresponding diffraction patterns they could play the same role as ordinary integers do in crystallography in this paper we prove interesting algebraic properties of the sets when is a quadratic unit pv number a class of algebraic integers which includes the quasicrystallographic cases we completely characterize their respective meyer additive and multiplicative properties where f and g are finite sets and also their respective galois conjugate sets these properties allow one to develop a notion of a quasiring we hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be module on a quasiring in we give also some two dimensional examples with

ulam stability problem for generalized a quadratic mappings abstract it is the purpose of this paper to investigate the stability problem of ulam for an approximate mapping for the following generalized quadratic functional equation i 1 n q j i x j n 1 x i n q i 1 n x i n 2 i 1 n q x i for all x i i 1 n where n 1

on the hyers ulam rassias stability of generalized quadratic mappings in banach modules we prove the generalized hyers ulam rassias stability of generalized a quadratic mappings of type p in banach modules over a banach algebra and of generalized a quadratic mappings of type r in banach modules over a banach algebra

the birth of e8 out of the spinors of the icosahedron e8 is prominent in mathematics and theoretical physics and is generally viewed as an exceptional symmetry in an eight dimensional 8d space very different from the space we inhabit for instance the lie group e8 features heavily in 10d superstring theory contrary to that point of view here we show that the e8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3d geometry the 240 roots of e8 arise in the 8d clifford algebra of 3d space as a double cover of the 120 elements of the icosahedral group generated by the root system h3 as a by product by restricting to even products of root vectors spinors in the 4d even subalgebra of the clifford algebra one can show that each 3d root system induces a root system in 4d which turn out to also be exactly the exceptional 4d root systems the spinorial point of view explains their existence as well as their unusual automorphism groups this spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions which opens up a novel interpretation of these phenomena in terms of spinorial geometry

voronoi and delaunay cells of root lattices classification of their faces and facets by coxeter dynkin diagrams the voronoi domains their duals delaunay domains and all their faces of any dimension are classified and described in terms of the weyl group action on a representative of each type of face the representative of a face type is specified by a decoration of the corresponding coxeter dynkin diagram the rules of domain description are uniform for root lattices of simple lie groups of all types an explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple lie groups the coxeter dynkin diagrams required here are the diagrams extended by the highest short root each diagram is partitioned into two subdiagrams one describing completely a d face of the voronoi domain its complement completely describing the dual of the d face the applicability of the authors classification method to generalized kaleidoscopes is explained

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

reduction of orbits of finite coxeter groups of non crystallographic type a reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices which transform points of the orbit of any group into points of the orbits of its subgroup projection matrices and branching rules for orbits of finite coxeter groups of non crystallographic type are presented the novelty in this paper is producing the branching rules that involve non crystallographic coxeter groups moreover these branching rules are relevant to any application of non crystallographic coxeter groups including molecular crystallography and encryption

classification of voronoi and delone tiles in quasicrystals i general method a new general method is presented which allows one to find all distinct voronoi and delone tiles in any quasicrystal from a large family this includes the tiles which may be present with arbitrarily low density 0 at all stages the method requires only consideration of a possibly large finite number of cases our method is applicable in principle to quasicrystals in any dimension and with any irrationality this is the first of three papers where the voronoi and delone tilings are studied two dimensional point sets quasicrystals arising from the a4 root lattice by means of the standard projection to a two dimensional plane with the irrationality 1 2 1 5 are considered in general we require that the acceptance window be bounded with non empty interior specific results are provided here for rhombic acceptance windows of any size oriented along the direction of simple roots of the coxeter group h2 within one quasicrystal the tiles are distinguished by their shape size and orientation the rhombic window case is indispensable for subsequent classification of voronoi and delone tiles in quasicrystals with general shape of the acceptance window voronoi and delone tiles of quasicrystals with circular and decagonal windows of any size are given in subsequent papers let vt denote the set of distinct voronoi tiles making up a quasicrystal with a given acceptance window there are three vt sets of the generic type and three of the singular type the latter occur for one precise value of the size of the acceptance window any other vt set is a uniform scaling of the tiles listed here similar results differing in detail are provided for the sets of distinct delone tiles dt altogether there are four different sets dt of delone tiles

a heuristic lagrangean algorithm for the capacitated plant location problem abstract lagrangean techniques have been widely applied to the uncapacitated plant location problem and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account in our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems the heuristics consisting of a two phase procedure the first location phase defines a set of multipliers from the analysis of the dual lp relaxation and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems several heuristics have been studied for this second phase based either on a decomposition of knapsack type subproblems through a definition of a set of penalties or of looking into the duality gap and trying to reduce it computational experience is reported

wavelets bases adapted to a self similar quasicrystal given any self similar quasicrystal in rn with inflation 1 we construct bases of l2 rn having the following structure nj 2 jx j z where the mother wavelets are smooth and with exponential decay or compact support we also show that wavelets constitute a relatively compact set in some sobolev space and that they depend continuously on when is equipped with an appropriate topology

hyers ulam stability of a generalized apollonius type quadratic mapping abstract let x y be linear spaces it is shown that if a mapping q x y satisfies the following functional equation 0 1 q i 1 n z i i 1 n x i q i 1 n z i i 1 n y i 1 2 q i 1 n x i i 1 n y i 2 q i 1 n z i i 1 n x i i 1 n y i 2 then the mapping q x y is quadratic we moreover prove the hyers ulam stability of the functional equation 0 1 in banach spaces

the problem of s m ulam for approximately multiplicative mappings in the present paper we study the problem of s m ulam for approximately multiplicative mappings and we consider new open problems

combining very large scale and ilp based neighborhoods for a two level location problem abstract in this paper we tackle a generalization of the single source capacitated facility location problem in which two sets of facilities called intermediate level and upper level facilities have to be located the dimensioning of the intermediate set the assignment of clients to intermediate level facilities and of intermediate level facilities to upper level facilities must be optimized as well such problem arises for instance in telecommunication network design in fact in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes which provide different services we propose a heuristic approach based on very large scale neighborhood search to tackle the problem in which both ad hoc algorithms and general purpose solvers are applied to explore the search space we report on experimental results using datasets from the capacitated location literature such results show that the approach is promising and that integer linear programming based neighborhoods are significantly effective

tau wavelets in the plane first an iterative process is described for subdivision and for scaling up of the penrose robinson tiles then we build the wavelets of haar i e suitably normalized characteristic functions on the tiles so that the orthogonality and multiresolution conditions are satisfied here 12 1 5

from cardinal spline wavelet bases to highly coherent dictionaries wavelet families arise by scaling and translations of a prototype function called the mother wavelet the construction of wavelet bases for cardinal spline spaces is generally carried out within the multi resolution analysis scheme thus the usual way of increasing the dimension of the multi resolution subspaces is by augmenting the scaling factor we show here that when working on a compact interval the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter by such a procedure we generate a redundant frame called a dictionary spanning the same spaces as a wavelet basis but with wavelets of broader support we characterize the correlation of the dictionary elements by measuring their coherence and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation

flows on hypergraphs we consider the capacitated minimum cost flow problem on directed hypergraphs we define spanning hypertrees so generalizing the spanning tree of a standard graph and show that like in the standard and in the generalized minimum cost flow problems a correspondence exists between bases and spanning hypertrees then we show that like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems most of the computations performed at each pivot operation have direct hypergraph interpretations

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

upper and lower bounds for the single source capacitated location problem abstract the single source capacitated location problem is considered given a set of potential locations and the plant capacities it must be decided where and how many plants must be open and which clients must be assigned to each open plant a lagrangean relaxation is used to obtain lower bounds for this problem upper bounds are given by lagrangean heuristics followed by search methods and by one tabu search metaheuristic computational experiments on different sets of problems are presented

the e8 lattice and quasicrystals geometry number theory and quasicrystals the authors study the quasiperiodic structures which can be derived from e8 this lattice suitably oriented leads to a 4d quasicrystal which has 3 3 5 symmetry they develop a modified version of the out and projection method the high dimensional lattice is first foliated into successive shells surrounding a vertex the shells are embedded into 7d s7 spheres they then use the so called hopf fibration of s7 to gather families of e8 shell sites into s3 fibres which contain 24 sites or a multiple of 24 which are symmetrically disposed in s3 this has two advantages first in the selection process a whole fibre is either selected or rejected which reduces the computation more importantly the selection process for a fiber amounts to simple arithmetical criteria the whole process leads to a shell by shell analysis of the 4d quasicrystal which can be brought onto a 2d 1d algorithm similar to the fibonacci chain construction they propose an arithmetic formula which gives the number of points on the shells this 4d quasicrystal has the interesting feature that it can be sliced into lower dimensional quasicrystals for example with icosahedral and tetrahedral symmetry in three dimensions

grand antiprism and quaternions vertices of the four dimensional 4d semi regular polytope the grand antiprism and its symmetry group of order 400 are represented in terms of quaternions with unit norm it follows from the icosian representation of the e8 root system which decomposes into two copies of the root system of h4 the symmetry of the grand antiprism is a maximal subgroup of the coxeter group w h4 it is the group aut h2 h 2 which is constructed in terms of 20 quaternionic roots of the coxeter diagram h2 h 2 the root system of h4 represented by the binary icosahedral group i of order 120 constitutes the regular 4d polytope 600 cell when its 20 quaternionic vertices corresponding to the roots of the diagram h2 h 2 are removed from the vertices of the 600 cell the remaining 100 quaternions constitute the vertices of the grand antiprism we give a detailed analysis of the construction of the cells of the grand antiprism in terms of quaternions the dual polytope of the grand antiprism has also been constructed

on the hyers ulam stability of linear mappings abstract let e1 be a real normed vector space and e2 a real banach space s m ulam posed the problem when does a linear mapping near an approximately linear mapping e1 e2 exist we give a new generalized solution to this problem an example illustrates when the answer to this question is negative the behaviour of bounded approximately additive mappings which do not satisfy hyers uiam stability is also investigated

tau wavelets of haar in this paper we construct a new type of haar wavelets called wavelets of haar using the arithmetics of the solutions and of the algebraic equation

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

a multi exchange heuristic for the single source capacitated facility location problem we present a very large scale neighborhood vlsn search algorithm for the capacitated facility location problem with single source constraints the neighborhood structures are induced by customer multi exchanges and by facility moves we consider both traditional single customer multi exchanges detected on a suitably definedcustomer improvement graph and more innovative multicustomer multi exchanges detected on afacility improvement graph dynamically built through the use of a greedy scheme computational results for some benchmark instances are reported that demonstrate the effectiveness of the approach for solving large scale problems a further test on real data involving an italian factory is also presented

a cut and solve based algorithm for the single source capacitated facility location problem abstract in this paper we present a cut and solve cs based exact algorithm for the single source capacitated facility location problem sscflp at each level of cs s branching tree it has only two nodes corresponding to the sparse problem sp and the dense problem dp respectively the sp whose solution space is relatively small with the values of some variables fixed to zero is solved to optimality by using a commercial mip solver and its solution if it exists provides an upper bound to the sscflp meanwhile the resolution of the lp of dp provides a lower bound for the sscflp a cutting plane method which combines the lifted cover inequalities and fenchel cutting planes to separate the 0 1 knapsack polytopes is applied to strengthen the lower bound of sscflp and that of dp these lower bounds are further tightened with a partial integrality strategy numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the cs approach in searching an optimal solution in a reasonable time computational results on large sized instances are also presented

multi exchange neighborhood structures for the capacitated minimum spanning tree problem the capacitated minimum spanning tree cmst problem is to find a minimum cost spanning tree with an additional cardinality constraint on the sizes of the subtrees incident to a given root node the cmst problem is an np complete problem and existing exact algorithms can solve only small size problems currently the best available heuristic procedures for the cmst problem are tabu search algorithms due to amberg et al and sharaiha et al these algorithms use two exchange neighborhood structures that are based on exchanging a single node or a set of nodes between two subtrees in this paper we generalize their neighborhood structures to allow exchanges of nodes among multiple subtrees simultaneously we refer to such neighborhood structures as multi exchange neighborhood structures our first multi exchange neighborhood structure allows exchanges of single nodes among several subtrees our second multi exchange neighborhood structure allows exchanges that involve multiple subtrees the size of each of these neighborhood structures grows exponentially with the problem size without any substantial increase in the computational times needed to find improved neighbors our approach which is based on the cyclic transfer neighborhood structure due to thompson and psaraftis and thompson and orlin transforms a profitable exchange into a negative cost subset disjoint cycle in a graph called an improvement graph and identifies these cycles using variants of shortest path label correcting algorithms our computational results with grasp and tabu search algorithms based on these neighborhood structures reveal that i for the unit demand case our algorithms obtained the best available solutions for all benchmark instances and improved some and ii for the heterogeneous demand case our algorithms improved the best available solutions for most of the benchmark instances with improvements by as much as 18 the running times our multi exchange neighborhood search algorithms are comparable to those taken by two exchange neighborhood search algorithms

on the generalized a quadratic mappings associated with the variance of a discrete type distribution abstract in this paper we prove the hyers ulam rassias stability of the following generalized quadratic functional equation of euler lagrange type i 1 n r i q j 1 n r j x i x j i 1 n r i q i 1 n r i x i i 1 n r i 2 i 1 n r i q x i in banach modules over a unital c algebra where n 1 is a positive integer and r i 0 i 1 n is arbitrary given

a repeated matching heuristic for the single source capacitated facility location problem facility location models form an important class of integer programming problems with application in many areas such as the distribution and transportation industries an important class of solution methods for these problems are so called lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust the general facility location problem can be divided into a number of special problems depending on the properties assumed in the capacitated location problem each facility has a specific capacity on the service it provides we describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility the approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied the method generates feasible solutions in each iteration in contrast to lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution numerical results show that the approach produces solutions which are of similar and often better than those produced using the best lagrangean heuristics

inflation centres of the cut and project quasicrystals a rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n dimensional euclidean space the inflation or equivalently the self similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete there are no other inflation centres for any chosen quasicrystal point internal inflation centre u its inflation properties are given as a set of scaling factors it turns out that the scaling factors form a one dimensional quasicrystal with a u dependent acceptance window scaling window the intersection of the scaling windows associated with all points of a quasicrystal is the one dimensional quasicrystal of universal internal scaling symmetries its acceptance window is the interval external inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres the scaling factors belonging to any such centre u form an infinite u dependent subset of points of the quasicrystal with acceptance window containing

a certain algebraic construction of quasicrystals and their isomorphism classes certain quasicrystals will be realized in cyclotomic fields and their isomorphism structures will be given in the case of seven fold or 30 fold symmetry

a heuristic for bilp problems the single source capacitated facility location problem abstract in the single source capacitated facility location problem sscflp each customer has to be assigned to one facility that supplies its whole demand the total demand of customers assigned to each facility cannot exceed its capacity an opening cost is associated with each facility and is paid if at least one customer is assigned to it the objective is to minimize the total cost of opening the facilities and supply all the customers in this paper we extend the kernel search heuristic framework to general binary integer linear programming bilp problems and apply it to the sscflp the heuristic is based on the solution to optimality of a sequence of subproblems where each subproblem is restricted to a subset of the decision variables the subsets of decision variables are constructed starting from the optimal values of the linear relaxation variants based on variable fixing are proposed to improve the efficiency of the kernel search framework the algorithms are tested on benchmark instances and new very large scale test problems computational results demonstrate the effectiveness of the approach the kernel search algorithm outperforms the best heuristics for the sscflp available in the literature it found the optimal solution for 165 out of the 170 instances with a proven optimum the error achieved in the remaining instances is negligible moreover it achieved on 100 new very large scale instances an average gap equal to 0 64 computed with respect to a lower bound or the optimum when available the variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality

variational problem of some quadratic functionals in complex analysis suppose we are given a pair of marked closed riemann surfacess twith the same genus we minimize the modified dirichlet integral for mappings ofsontot which is endowed with some weight in terms of conformal metric on the target surfacet in this paper it is demonstrated that the minimal harmonic map exists and it proves to be teichmuller

affine extension of noncrystallographic coxeter groups and quasicrystals unique affine extensions haff2 haff3 and haff4 are determined for the noncrystallographic coxeter groups h2 h3 and h4 they are used for the construction of new mathematical models for quasicrystal fragments with tenfold symmetry the case of haff2 corresponding to planar point sets is discussed in detail in contrast to the cut and project scheme we obtain by construction finite point sets which grow with a model specific growth parameter

an exact algorithm for the capacitated facility location problems with single sourcing facility location problems are often encountered in many areas such as distribution transportation and telecommunication we describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility an important class of heuristic solution methods for these problems are lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust a primal heuristic based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied is incorporated into the lagrangian heuristic finally a branch and bound method based on the lagrangian heuristic is developed and compared computationally to the commercial code cplex the computational results indicate that the proposed method is very efficient

root lattices and quasicrystals it is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models all noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry

a highly symmetric four dimensional quasicrystal a quasiperiodic pattern or quasicrystal is constructed in real four dimensional euclidean space having the non crystallographic reflection group 3 3 5 of order 14 400 as its point group it is obtained as a projection of the eight dimensional lattice e8 and has as a cross section a three dimensional quasicrystal with icosahedral symmetry

a hypergraph multi exchange heuristic for the single source capacitated facility location problem abstract in this paper we introduce a large scale neighborhood search procedure for solving the single source capacitated facility location problem sscflp the neighborhood structures are induced by innovative split multi customer multi exchanges where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa to represent these exchanges we use two types of improvement hypergraphs the improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed integer program we develop a hypergraph search framework including forward and backward procedures to identify improving solutions efficiently our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi exchange heuristic ahuja orlin pallottino scuttella 2004 in addition when compared with the kernel search algorithm guastaroba speranza 2014 which at present is the most effective for solving sscflp our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

wavelet multiresolutions for the fibonacci chain most of the discrete wavelet families are periodic dyadic they live on the multiresolution sequence 2j j of lattice backgrounds at different scales we present here a construction of wavelet family based on a sequence of aperiodic discretizations of at a given scale this discretization is the point set of nodes of the fibonacci chain a well known stone inflation cut and project model for one dimensional quasicrystals corresponding multiresolution analysis and the elementary example of the haar system are presented

densities minimal distances and coverings of quasicrystals quasicrystals are generalizaions of lattices in this spirit we consider here three kinds of geometric properties of mathematical models of quasicrystals minimal distances between quasicrystal points covering radii of quasicrystals and the densities of quasicrystal points

proving the deterministic period breaking of linear congruential generators using two tile quasicrystals we describe the design of a family of aperiodic prngs aprngs we show how a one dimensional two tile cut and project quasicrystal 2tqc used in conjunction with lcgs in an aprng generates an infinite aperiodic pseudorandom sequence in the suggested design any 2tqc corresponding to unitary quadratic pisot number combined with either one or two different lcgs can be used

asymptotic aspect of the quadratic functional equation in multi normed spaces abstract we investigate the hyers ulam stability of the quadratic functional equation for mappings from abelian groups into multi normed spaces we also study the stability on a restricted domain and present an asymptotic behavior of the quadratic equation in the framework of multi normed spaces

generating functions for the coxeter group generating functions are found for decomposition of the space of homogeneous polynomials of any degree in four variables into the direct sum of subspaces irreducible under the group the non crystallographic coxeter group of order 14 400 the four variables are coordinates of a vector from the defining four dimensional representation space of the group as the defining representation we consider any of the four non equivalent irreducible representations of of dimension four analogous generating functions for the binary icosahedral group of order 120 generated by reflections i e a subgroup of but not of and for the dihedral group of order 10 are also rederived and shown the groups are naturally related by