penalty functions in e programming and e minimax problems several authors have been intersted in optimality conditions in e programming and in e minimas problem see for example the references 11 16

self similarity of quasilattices in two dimensions ii the non bravais type n gonal quasilattice for pt i see ibid vol 22 p 193 1989 presents a systematic method of dividing an n gonal lattice into identical sublattices which are also n gonal lattices a non bravais type n gonal quasilattice in two dimensions is constructed with the projection method by assigning windows with different shapes sizes and or orientations to the sublattices the symmetries and the orientations of the windows are determined by the relative point symmetries of the sublattices to the n gonal lattice and windows assigned to equivalent sublattices differ from one another only in their orientations the sublattices are transformed among themselves by a volume conserving linear transformation induced by a complex pv unit in the n cyclotomic field a necessary and sufficient condition for a pv unit to be a complex self similarity ratio of a non bravais type quasilattice is presented it is shown that every non bravais type n gonal quasilattice has a self similarity characterised by pv unit

the space groups of orthorhombic approximants to the icosahedral quasilattice the space groups of the orthorhombic approximant lattices to the primitive icosahedral quasilattice are classified there exist three bravais classes pmmm cmmm and immm the basis vectors of the bravais lattice of an approximant are parallel to two three and or fivefold axes of the quasilattice it is found that there exist many nonsymmorphic space groups with a common bravais lattice in addition to symmorphic ones this is because glides commonly appear

e optimality and duality for multiobjective fractional programming abstract using the scalar e parametric approach we establish the karush kuhn tucker which we call kkt necessary and sufficient conditions for an e pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints these optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems subsequently we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem under this case we derive the kkt necessary and sufficient conditions without a constraint qualification for e pareto optimality of multiobjective fractional programming

evaluation of the undetected error probability of single parity check product codes in this paper an expression for the undetected error probability p epsilon of single parity check product spcp codes used for error detection over a binary symmetric channel is derived it is shown that square spcp codes need not obey a certain commonly used bound approximate expressions for the maximum p epsilon and the corresponding maximizing e are given

poor error correction codes are poor error detection codes corresp we show that binary group codes that do not satisfy the asymptotic varshamov gilbert bound have an undesirable characteristic when used as error detection codes for transmission over the binary symmetric channel

epsilon approximate solutions for multiobjective programming problems abstract in this paper approximate solutions for multiobjective programming problems up to 0 are investigated the concepts of approximate solutions were introduced by white and loridan the connections with these concepts are studied also the sufficient and necessary conditions to obtain the elements of the smallest approximate solution set are verified

theory of self similarity of periodic approximants to a quasilattice it is shown that self similarity of a quasilattice has a profound effect on the periodic approximants to the quasilattice the periodic approximants are grouped into series so that i each series is generated from its prototype by a successive application of the deflation and rescaling ii the space group is common among the members of the series and iii the unit cell of the approximant is scaled up by tau with the series number where tau is the scale of self similarity of the relevant quasilattice these results are exemplified with application to the case of the octagonal quasilattice

special points of 2 1 reducible quasilattices in three dimensions presents a complete classification of special points of five dimensional 5d bravais lattices which yield with the cut and projection method 2 1 reducible quasilattices in 3d the quasilattices are periodic along the c axis but quasiperiodic only along the plane perpendicular to it there exist five bravais classes of 5d lattices associated with the 2 1 reducible quasilattices namely primitive octagonal decagonal and dodecagonal lattices the centred octagonal lattice and the pentagonal lattice the author also discusses the special points of the reciprocal lattices of these 5d lattices

ϵ pareto optimality for nondifferentiable multiobjective programming via penalty function abstract necessary and sufficient conditions without a constraint qualification for pareto optimality of multiobjective programming are derived the necessary kuhn tucker condition suggests the establishment of a wolf type duality theorem for nondifferentiable convex multiobjective minimization problems the generalized saddle point for pareto optimality of the vector lagrangian is studied

special lines of quasilattices i the case of irreducible quasilattices in two and three dimensions a general theory for special lines in quasilattices is developed using the theory the author classifies completely the special lines of the three bravais classes of icosahedral quasilattices in three dimensions and another three of n gonal quasilattices in two dimensions where n 8 10 and 12 he establishes compatibility relationships between the classes of special lines and those of special points for each of the six bravais classes of quasilattices

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

composite nonsmooth multiobjective programs with v ρ invexity abstract in this paper some problems consisting of nonsmooth composite multiobjective programs have been treated with v invexity type of conditions in particular we prove the generalized karush kuhn tucker sufficient optimality theorem and duality theorems for nonsmooth composite multiobjective programs also weak vector saddle point theorems are obtained for the composite programs under v invexity conditions the results obtained generalize the results of hun kuk et al to some extent

theory of self similarity of periodic approximants to a quasilattice ii the case of the dodecagonal quasilattice for pt i see ibid vol 24 p 4873 87 1991 the space groups of the periodic approximants pas to the dodecagonal quasilattice in two dimensions are investigated the bravais lattices of the pas are tetragonal p4mm rectangular pmm hexagonal p6mm and rhombic cmm the author presents several series of pas each of which is derived from a prototype pa by a successive application of the deflation and rescaling dar and the space group is common among the members each series is composed of two subseries the consecutive members in each subseries are related by the proper dar and the two subseries are related to each other by the improper dar

generating epsilon efficient solutions in multiobjective programming scalarization approaches to purposely generating e efficient solutions of multiobjective programs are investigated and a generic procedure for computing these solutions is proposed and illustrated with an example real life decision making situations in which the solutions are of significance are described

a classification of special points of quasilattices in two dimensions a complete classification of the special points of octagonal decagonal and dodecagonal quasilattices in two dimensions 2d is presented they are obtained by projecting the special points of the starting n gonal lattice n 8 10 or 12 in 4d onto a 2d subspace a set of equivalent special points of a quasilattice are located on the centres of a single kind of tile in the quasiperiodic tiling associated with the quasilattice the point symmetry of the special points is identical to that of the tiles they form a quasilattice with the same point symmetry as that of the original quasilattice the new quasilattice is of a bravais type or non bravais type according to whether the point symmetry group of the special points is identical to the full point symmetry group of the quasilattice or its true subgroup respectively

e optimality for multiobjective programming on a banach space abstract we establish a lagrange multiplier rule in terms of clarke generalized gradient that characterizes e efficiency of a nondifferentiable multiobjective programming problem on a real banach space this rule is then utilized to establish relationship between e efficient solution of multiobjective program and generalized saddle point of an appropriate vector valued lagrangian under invexity hypothesis

a classification of special points of icosahedral quasilattices investigates special points of icosahedral quasilattices in three dimensions 3d they are given as the projections onto the real space of the special points of icosahedral lattices which are periodic lattices in 6d there exist three bravais classes of 6d icosahedral lattices and the author present a complete classification of their special points

theory of self similarity of periodic approximants to a quasilattice iii the case of a nonbravais type quasilattice for pt ii see ibid vol 25 p 2147 1992 the mother lattice l of a non bravais type quasilattice nbtql is a non bravais type periodic lattice with higher dimensionality and has an associated bravais lattice l0 the main problems on periodic approximants pas to the nbtql are reduced to similar problems in the case of the relevant bravais type quasilattice derived from l0 and the previous theories on the space groups and self similarity of the pas apply to the nbtql the present theory includes a general prescription of obtaining pas to a nbtql the author applies the theory to several nbtqls with an octagonal decagonal or dodecagonal point symmetry the second important bravais lattice lh called the host lattice is associated with l and he has clarified the difference in roles between l0 and lh in the theory of pas to the nbtql

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

two dimensional dodecagonal quasilattices two dimensional quasiperiodic lattices with 12 fold rotational symmetry are constructed by projecting a four dimensional hyperhexagonal lattice onto a two dimensional subspace the projection method in the present case is greatly simplified by representing the four dimensional lattice as a complex two dimensional lattice by choosing windows with different forms one obtains several quasiperiodic tilings which have a self similarity one of the tilings is quite similar to the pattern of the dodecagonal quasicrystal of a nicr alloy

a chain rule for ɛ subdifferentials with applications to approximate solutions in convex pareto problems abstract in this work we obtain a chain rule for the approximate subdifferential considering a vector valued proper convex function and its post composition with a proper convex function of several variables nondecreasing in the sense of the pareto order we derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula to get the results we require qualification conditions since in the composition the initial function is extended vector valued this chain rule extends analogous well known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition as application we deduce the well known calculus rule for the addition and we extend the formula for the maximum of functions finally we use them and a scalarization process to obtain kuhn tucker type necessary and sufficient conditions for approximate solutions in convex pareto problems these conditions extend other obtained in scalar optimization problems

an n gonal quasilattice in two dimensions as a dual lattice to a multiple periodic grid an n gonal quasilattice in two dimensions is constructed for the case of n 6m m or 2 as a dual to an m tuple periodic grid gm g0 union rg0 union union rm 1g0 where g0 is a periodic grid with hexagonal point symmetry and r the rotation operation of the plane by angle 2 pi n the author investigates the case where g0 is a honeycomb triangular kagome or diced grid the resulting quasilattice is not of bravais type unless g0 is the honeycomb grid it is shown also that the same quasilattice is obtained with the projection method from the 2m dimensional lattice l l0 l0 l0 where l0 is a triangular honeycomb diced or kagome lattice being dual to g0 in the projection method the 2m dimensional euclidean space is divided into cells with a 2m dimensional grid g which is dual to l it is shown that gm is identical to a two dimensional section of g

a classification of two dimensional quasiperiodic tilings obtained with the grid method two dimensional quasiperiodic tilings qpt obtained from n grids are classified into local isomorphism li classes by using the invariants of the grids the number of the invariants is given by n phi n if n is odd or n phi 2n if n is even where phi is euler s function in number theory all the qpt obtained from n grids with n 2k k or 2 belong to a single li class whose point symmetry is d2n when n is not a power of 2 the qpt are classified into several li classes of the li classes two have the highest point symmetry d2n one of the two classes is associated with n grids in which all the invariants vanish and the other with n grids in which all of them take 1 2 in the case of an odd n there is also one continuous series of qpt with point symmetry dn the author also presents a general formula of the tile statistics of the quasiperiodic tilings obtained with the grid method

ϵ optimality criteria for vector minimization problems via exact penalty functions abstract several authors have been interested in optimality criteria for scalar programming problems in this paper some optimality results for the scalar problems of ours math programming 56 1992 233 243 are extended to vector minimization problems via exact penalty functions similar to our paper the solvability of neither the vector problems nor the associated dual problems is assumed our criteria are derived by estimating the size of the penalty parameter in terms of approximate solutions for the dual problems

special lines of quasilattices ii the case of 2 1 reducible quasilattices in three dimensions for pt i see ibid vol 24 p 1023 1991 the author presents a complete classification of special lines from 2 1 reducible quasilattices which are periodic along the c axis but only quasiperiodic along the plane perpendicular to it he also presents compatibility relationships between the classes of special lines and those of the special points

e optimality criteria for convex programming problems via exact penalty functions in this paper we presente optimality criteria for convex programming problems associated with exact penalty functions several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved we assume the solvability of neither the convex problem nor the dual problem to derive our criteria we estimate the size of the penalty parameter in terms of ane solution for the dual problem

three dimensional black and white bravais quasilattices with 2 1 reducible point groups the author has investigated three dimensional black and white bravais quasilattices bwbql with point groups d8h 8 mmm d10h 10 mmm d12h 12 mm4im and d5d 5 m which are 2 1 reducible there exists seven bravais classes of bwbqls p2c8 mmm pp8 mmm pf8 mmm fp8 mmm p2c10 mmm p2c12 mmm and pp5m

self similarity of quasilattices in two dimensions iii inflation by a non unit pv number for pt ii see ibid vol 22 205 1989 it is shown that every n gonal quasilattice in two dimensions has self similarity with a complex self similarity ratio which is equal to a non unit pv number in the n cyclotomic field in this self similarity the original quasilattice can be inflated to a number of different quasilattices which are locally isomorphous to each other the number is equal to the algebraic norm of the relevant pv number in the n cyclotomic field several examples are presented for the cases of octagonal decagonal and dodecagonal quasilattices the difference between the present self similarity and that associated with a pv unit is clarified

the torus parametrization of quasiperiodic li classes the torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties the method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling namely for the fibonacci chain one dimensional the triangle and penrose patterns two dimensional and for kramer s and danzer s icosahedral tilings three dimensional we obtain a rather complete picture of the orbit structure within these classes and also discuss various general results

a classification of the space groups of approximant lattices to a decagonal quasilattice a periodic approximant to a two dimensional 2d decagonal quasilattice is obtained by the projection method from a 4d periodic lattice l which is a commensurate deformation of the 4d decagonal lattice the bravais lattice of the approximant is given by the restriction of l onto the physical space while its space group by the symmetry of the phase vector with respect to the 2d lattice ls which is the projection of l onto the internal space there exist 12 space groups of the rectangular approximants pmm pmg pgm pgg pm1 p1m pg1 p1g cmm gamma cmm y cm1 and c1m which are derived from the special points or special lines of ls

self similarity of quasilattices in two dimensions i the n gonal quasilattice the self similarity of a two dimensional quasilattice with an n gonal point symmetry has been investigated for even n n or 8 it is shown that for any n an n gonal quasilattice has a self similarity characterised by a complex number tau on inflation the quasilattice is scaled by mod tau mod and subsequently rotated by arg tau tau is a pv unit of the n cyclotomic field q zeta zeta exp 2 pi i n tau satisfies i tau and tau 1 are both algebraic integers in q zeta ii mod tau mod 1 and iii mod tau mod 1 for any conjugate tau but tau the complex conjugate of tau in q zeta pv units are calculated for every n gonal quasilattice whose multiplicity m phi n 2 with phi being the eulerian function is less than 5 m 2 for n 8 10 12 m 3 for n 14 18 and m 4 for n 16 20 24 30 it is found that a quasilattice has two or more independent scales of self similarity if its multiplicity is larger than 2

a dual differentiable exact penalty function a new penalty function is associated with an inequality constrained nonlinear programming problem via its dual this penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable in addition the penalty parameter remains finite this approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a euclidean space and the nonnegative orthant of another euclidean space many efficient algorithms exist for solving this problem for the case of quadratic programming the penalty function problem can be solved effectively by successive overrelaxation sor methods which can handle huge problems while preserving sparsity features

equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings the two main techniques for the generation of quasiperiodic tilings de bruijn s grid method 1981 and the projection formalism are generalised a very broad class of quasiperiodic tilings is obtained in this way the two generalised methods are shown to be equivalent the standard calculation of fourier spectra is extended to the whole general class of tilings various applications are discussed

e optimal solutions in nondifferentiable convex programming and some related questions in this paper we present e optimality conditions of the kuhn tucker type for points which are within e of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints such e optimality conditions are of interest for theoretical consideration as well as from the computational point of view some illustrative applications are made thus we derive an expression for the e subdifferential of a general convex max function we also show how the e optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints

a self similar dodecagonal quasiperiodic tiling of the plane in terms of squares regular hexagons and thin rhombi the author presents a dodecagonal quasiperiodic tiling of the plane in terms of three kinds of tiles a square a regular hexagon and a thin rhombus whose acute inner angles are 30 degrees the dodecagonal tiling is self similar with respect to a dilatation by 2 square root 3 it is associated with a dodecagonal quasiperiodic lattice which is obtained by a projection from a hyperhoneycomb lattice in four dimensions the hyperhoneycomb lattice is a non bravais lattice composed of four bravais sublattices and the author must assign windows with different orientations to different sublattices the author discusses in detail the relationships of the present method of obtaining a dodecagonal tiling to other methods

self similar quasilattices with windows having fractal boundaries the point inflation rule is presented as a powerful scheme to produce a self similar quasilattice with a non crystallographic point symmetry alternatively the conjugate point inflation rule which is a set map acting on the internal space has the unique fixed point which is a compact set w and the quasilattice ql is obtained with the projection method by taking w as the window the fixed point w is the attractor of the set map if w is topologically a disc a ball for a three dimensional case its boundary is usually fractal we investigate conditions for the boundary of the window to be a von koch curve a practical method of constructing windows and the relevant qls is presented a lot of windows with fractal boundaries are determined on the basis of this method we introduce isomerism which has priority of rank to the mutual local derivability as a classification scheme for self similar qls it is argued that the pir and isomerism can be the key points of classification of self similar qls

more results on the weight enumerator of product codes we consider the product code c sub p of q ary linear codes with minimum distances d sub c and d sub r the words in c sub p of weight less than d sub r d sub c max d sub r spl lceil d sub c g spl rceil d sub c spl lceil d sub r q spl rceil are characterized and their number is expressed in the number of low weight words of the constituent codes for binary product codes we give an upper bound on the number of words in c sub p of weightless than min d sub r d sub c spl lceil d sub c 2 spl rceil 1 d sub c d sub r spl lceil d sub r 2 spl rceil 1 that is met with equality if c sub c and c sub r are extended perfect codes

on testing for improper error detection codes a family of tests for improper codes is given these tests can be used in cases where the complete weight distribution of the code is unknown it was found that knowledge of the number of minimum weight codewords can be used to greatly increase the effectiveness of the asymptotic varshamov gilbert test further improvement is possible as more is known about the number of other weight codewords

sufficiency of exact penalty minimization by employing a recently obtained error bound for differentiable convex inequalities it is shown that under appropriate constraint qualifications a minimum solution of an exact penalty function for a single value of the penalty parameter which exceeds a certain threshold is also a solution of the convex program associated with the penalty function no a priori assumption is made regarding the solvability of the convex program if such a solvability assumption is made then we show that a threshold value of the penalty parameter can be used which is smaller than both the above mentioned value and that of zangwill these various threshold values of the penalty parameter also apply to the well known big m method of linear programming

a note upon another error detecting code that is not proper in this correspondence the performance of the nonlinear error detecting code described in ccir recommendation 476 2 when used over a memoryless binary symmetric channel bsc is derived from this it is apparent that the probability of accepting an erroneous block is not a monotonic function of the bit error probability but rather reaches a peak of 5 653 percent when the bit error probability is 11 66 percent consequently this code is not proper 1 6