alternative characterizations for the representation of families of sets by fuzzy sets we investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets this concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set negoita and ralescu obtained two theorems both giving necessary and sufficient conditions one called the representation theorem and the second called the generalized representation theorem our aim in this paper is to take a closer look at both theorems hence establishing some equivalent forms and several constructive remarks including some directions for future investigations

operator approach to performance analysis of root music and root min norm the authors carry out a performance analysis of two eigenstructure based direction of arrival estimation algorithms using a series expansion of projection operators or projectors on the signal and noise subspaces in the interest of algebraic simplicity an operator formalism is utilized a perturbation analysis is performed on the projectors the results of which are used to determine the effect on the estimated parameters the approach makes it possible to carry out the analysis to any chosen order of expansion of the projectors by using an original recurrence formula developed for the higher order terms in the series expansion of the projectors this method is used to study the root music and root min norm algorithms and establish the superiority of root music in all cases the analysis has also resulted in insightful asymptotic expressions that describe the statistical behavior of the estimated angles and radii of the signal zeros

flocking of multi agent systems with multiplicative and independent measurement noises abstract we investigate the effect of noise and group density on the emergence of flocking in a stochastic cucker smale type system we consider the independent and multiplicative measurement noise and derive sufficient conditions for flocking and non flocking in terms of the noise intensity and group density we report that in noise environments it is rather easy for the group with high density to form and maintain ordered group motion moreover we find that under the perturbation of multiplicative noise systems with weak coupling can tolerate relatively strong perturbation of noise

cucker smale flocking under hierarchical leadership a mathematical theory on flocking serves the foundation for several ubiquitous multiagent phenomena in biology ecology sensor networks and economics as well as social behavior like language emergence and evolution directly inspired by the recent fundamental works of cucker and smale on the construction and analysis of a generic flocking model we study the emergent behavior of cucker smale flocking under hierarchical leadership the rates of convergence towards asymptotically coherent group patterns in different scenarios are established the consistent convergence towards coherent patterns may well reveal the advantages and necessities of having leaders and leadership in a complex biological technological economic or social system with sufficient intelligence

existence and uniqueness of fuzzy ideals let r be a commutative ring the idea of identifying a fuzzy ideal of r with its corresponding chain of level ideals and its range of values although widely used in the literature for the study of fuzzy ideals has never been investigated in a general setting before we investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established in particular we obtain that uniqueness always holds true for artinian rings moreover examples are presented to further support our results and shed more light on the above mentioned problem of identification we finish by raising a natural question that is left as an open problem for further investigation

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

hierarchical cucker smale model subject to random failure consider a system of autonomous interacting agents in r3 where each agent adjusts its velocity according to its own past velocity and a weighted mean of the relative velocities of its superiors in a certain prescribed hierarchy furthermore each agent at each time step can fail to see any of its superiors with a certain probability the failure rate probability we prove that for agent interactions with linear or sub linear distance dependent rate of decay the flocking results valid for the non failure deterministic model introduced by shen also hold true independently of the failure rate probability

flocking estimates for the cucker smale model with time lag and hierarchical leadership abstract we analyze the cucker smale model under hierarchical leadership in presence of a time delay by using a lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay we also prove a flocking estimate in the case of a free will leader these results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects that frequently appear in real situations

representing ordered structures by fuzzy sets an overview we present a survey on representations of ordered structures by fuzzy sets any poset satisfying some finiteness condition semilattice lattice belonging to a special class e g distributive noetherian complete and others can be represented by a single function i e by a fuzzy set its domain and co domain are particular subsets of the same structure and consist of irreducible elements the representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by its proper subset by this approach the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion

a sensitivity analysis of the music algorithm a first order analysis of the sensitivity of the music algorithm to system errors which cause differences between the array manifold used by music and the true array manifold is presented the effect of such errors on the direction of arrival estimates is investigated the ability of the music algorithm to resolve two closely spaced sources in the presence of phase and gain errors in the array elements or the receivers or errors in the element locations is analyzed a formula for computing the failure threshold of music as a function of source separation and other system parameters is derived and tested by simulation the analysis assumes that the exact covariance matrix of array element outputs is known the infinite data case

on the uniqueness of l fuzzy sets in the representation of families of sets abstract this paper deals with the uniqueness of l fuzzy sets in the representation of a given family of subsets of a nonempty set it first shows a formula of the number of l fuzzy sets whose collection of cuts coincides with a given family of subsets of a nonempty set and then provides a necessary and sufficient condition under which such l fuzzy set is unique

direction finding for diversely polarized signals using polynomial rooting a direction finding algorithm for diversely polarized arrays that is based on polynomial rooting is presented using polynomial rooting instead of search reduces significantly the computational requirements of the algorithm and enhances resolution the basic method is limited to linear uniformly spaced arrays however using an interpolation technique the algorithm is extended to a larger class of arrays the performance of the proposed algorithm is analyzed computer simulations are used to demonstrate the performance of the new technique and to verify the performance analysis

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

a generalization of the representation theorem abstract we generalize the representation theorem for fuzzy sets more specifically we give necessary and sufficient conditions for a family of sets to represent a permutation of the levels of some fuzzy set

emergent behavior of a cucker smale type particle model with nonlinear velocity couplings in this note we present a cucker smale type flocking model with nonlinear velocity couplings and derive sufficient conditions for the formation of flocking in terms of communication weight and initial spatial velocity standard deviations

a general collision avoiding flocking framework we prove a general result of collision avoiding flocking the underlying model allows several forms of coupling forces and the main result ensures flocking provided the initial state of the population does not show simultaneously very different velocities very spread positions or very close agents

uniqueness results in the representation of families of sets by fuzzy sets the important concept of identifying a fuzzy subset of a set r with its corresponding chain of level sets is widely used in the literature we investigate in this paper the problem of characterization of all fuzzy subsets of r that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 although our main focus is on the problem of uniqueness we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets we continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed to further support our results we present some examples which we hope will shed more light on the above mentioned problem of identification we finish by raising a few natural questions that are left as open problems for further investigation

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

representation of lattices by fuzzy sets abstract we prove that every lattice l of finite length can be represented by a fuzzy set on the collection x of meet irreducible elements of l a decomposition of this fuzzy set gives a family of isotone functions from x to 2 0 1 the lattice of which is isomorphic to l more generally conditions under which any collection of isotone functions from a finite set into 2 corresponds to a decomposition of a fuzzy set are given as a consequence the representation theorem for a finite distributive lattice by the lattice of all isotone functions is obtained the collection of all lattices characterized by the same fuzzy set turns out to be a lattice with the above mentioned distributive lattice as the greatest element

a complete characterization of extinction versus positivity of solutions to a parabolic problem of p laplacian type in graphs abstract this work is concerned with the extinction and positivity properties of solutions to the discrete p laplacian parabolic equation u t p u f u p 1 with a nonlinear absorption on a weighted graph we provide a complete characterization for f to see when we have an extinctive solution or a positive solution in addition the extinction time is estimated for the extinctive solutions and the decay rate is given to the positive solutions finally we give several numerical examples which illustrate the main results

homomorphisms of poset valued algebras the notion of a homomorphism for fuzzy algebraic structures having arbitrary possibly different partially ordered sets or lattices as co domains is introduced basic properties of homomorphisms embeddings and isomorphisms are investigated and fundamental homomorphism and isomorphism theorems are proved

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

cucker smale flocking under rooted leadership with free will agents abstract motivated by the recent works of cucker smale and shen we study the emergent behavior of discrete time cucker smale flocking under rooted leadership with free will agents we prove that if the free will accelerations of the agents relative to the leader are summable then the individual velocities will converge to the velocity of leader under suitable initial conditions in particular we prove that the unconditional convergence is true if the decay exponent of the communication weight is below some critical value which depends on the interaction topology we also present some numerical examples to illustrate the main result

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

second order consensus of multi agent systems under limited interaction ranges this paper studies a distributed linear consensus protocol for second order multi agent systems under limited agent interaction ranges in particular two agents can interact with each other only if their distance is within a certain range under a linear consensus protocol with the relative state feedback we derive a sufficient condition on the initial states and the interaction range to achieve the second order consensus by using a lyapunov functional approach finally simulation examples are included to validate the theoretical result

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

flocking with short range interactions we study the large time behavior of continuum alignment dynamics based on cucker smale cs type interactions which involve short range kernels that is communication kernels with support much smaller than the diameter of the crowd we show that if the amplitude of the interactions is larger than a finite threshold then unconditional hydrodynamic flocking follows since we do not impose any regularity nor do we require the kernels to be bounded the result covers both regular and singular interaction kernels moreover we treat initial densities in the general class of compactly supported measures which are required to have positive mass on average over balls at small enough scale but otherwise vacuum is allowed at smaller scales consequently our arguments of hydrodynamic flocking apply mutatis mutandis to the agent based cs model with finitely many dirac masses in particular discrete flocking threshold is shown to depend on the number of dense clusters of communication but otherwise does not grow with the number of agents

the resolution threshold of a direction finding algorithm for diversely polarized arrays subspace based algorithms for narrowband direction of arrival doa estimation require detailed knowledge of the array response the array manifold and assume that the noise covariance matrix is known up to a scaling factor in practice these quantities are not known precisely resolution and estimation accuracy can degrade significantly when the array response or the noise covariance deviate from their nominal values we examine the resolution threshold of a recently proposed subspace based algorithm for direction finding with diversely polarized arrays we study finite sample effects and the effects of modeling errors errors in the array manifold or the noise covariance on the resolution threshold a comparison is made between the resolution thresholds of the music algorithm for uniformly polarized arrays and the proposed algorithm for diversely polarized arrays

completion of ordered structures by cuts of fuzzy sets an overview the aim of the paper is to present a role of fuzzy sets in the theory of ordered structures main algebraic properties of cuts of fuzzy sets are given and a completion of partially ordered sets to complete lattices is described it turns out that this completion is equivalent with the famous dedekind macneille completion but the algorithm presented here is much simpler

asymptotic behavior and collision avoidance in the cucker smale model for the long range communicated cucker smale c s model asymptotic flocking can be obtained by taking full use of a key equality about the kinetic energy however such behavior does not exist for the short range communicated c s model with general initial data in this note we first establish a new equality about the second order velocity position moments for general c s models then we use this equality to deduce a new type of asymptotic behavior for the short range communicated c s model such behavior not only reflects an essential feature of the short range communication weight but also has some potential applications in the complex multicluster problem besides collision avoidance is also obtained for the c s model with some singular short range communication weights

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

emergent behavior in flocks we provide a model for both continuous and discrete time describing the evolution of a flock our model is parameterized by a constant beta capturing the rate of decay which in our model is polynomial of the influence between birds in the flock as they separate in space our main result shows that when beta 1 2 convergence of the flock to a common velocity is guaranteed while for betages1 2 convergence is guaranteed under some condition on the initial positions and velocities of the birds only

on a representation of posets by fuzzy sets the aim of the paper is to present an application of fuzzy sets in mathematics namely in the theory of ordered sets an algorithm for the construction of p fuzzy sets with distinct levels is given in connection with this every finite poset can be mapped by an anti isotone bijection onto the poset of levels of a suitable fuzzy set moreover it is proved that every finite partially ordered set p can be represented by the poset of levels of a particular fuzzy set defined on the collection of meet irreducible elements of p