all degree 6 local unitary invariants of k qudits we give explicit index free formulae for all the degree 6 and also degree 4 and 2 algebraically independent local unitary invariant polynomials for finite dimensional k partite pure and mixed quantum states we carry this out using graph technical methods which provide illustrations for this abstract topic

global existence to the vlasov poisson system and propagation of moments without assumption of finite kinetic energy we consider classical as well as weak solutions to the three dimensional vlasov poisson system without assuming finiteness of kinetic energy we prove global existence of classical solutions by assuming the initial datum is smooth enough and has a compact velocity spatial support which will be specified in theorem 1 1 we also establish some propagation results for low moments of weak solutions

flocking of the cucker smale model on general digraphs in the literature the study of the cucker smale c s flocking model is all restricted to connected undirected graphs or digraphs of some special types containing only a single root in this technical note we establish the flocking behavior of the c s model on general digraphs which contain all existing interaction topologies in relation to the c s model as special cases in particular the topology with multiple leaders is included the flocking results are guaranteed under some conditions on the initial state of the flock only the critical exponent below which unconditional convergence holds is given depending only on the interaction topology it is consistent with the ones obtained under some known special topologies in the literature

three qubit pure state canonical forms in this paper we analyse the canonical forms into which any pure three qubit state can be cast the minimal forms i e the ones with the minimal number of product states built from local bases are also presented and lead to a complete classification of pure three qubit states this classification is related to the values of the polynomial invariants under local unitary transformations by a one to one correspondence

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

avoiding collisions in flocks among the many models for flocking systems of interacting particles the one introduced by cucker and smale has attracted attention due to the fact that a convergence to flocking i e to a common velocity could be established depending on conditions on the initial state of the system in this note we extend this model by adding to it a repelling force between particles we show that for this modified model convergence to flocking is established along the same lines while in addition avoidance of collisions i e the respect of a minimal distance between particles is ensured

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

multipartite generalization of the schmidt decomposition we find a canonical form for pure states of a general multipartite system in which the constraints on the coordinates with respect to a factorizable orthonormal basis are simply that certain ones vanish and certain others are real for identical particles they are invariant under permutations of the particles as an application we find the dimension of the generic local equivalence class

existence of piecewise weak solutions of a discrete cucker smale s flocking model with a singular communication weight abstract we prove existence of global c 1 piecewise weak solutions for the discrete cucker smale s flocking model with a non lipschitz communication weight s s 0 1 we also discuss the possibility of finite in time alignment of the velocities of the particles

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

on local invariants of pure three qubit states we study invariants of three qubit states under local unitary transformations i e functions on the space of entanglement types which is known to have dimension six we show that there is no set of six algebraically independent polynomial invariants of degree 6 and find such a set with maximum degree eight we describe an intrinsic definition of a canonical state on each orbit and discuss the non polynomial invariants associated with it

a complete set of local invariants for a family of multipartite mixed states we study the equivalence of quantum states under local unitary transformations by using the singular value decomposition a complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices

generalized schmidt decomposition and classification of three quantum bit states we prove for any pure three quantum bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form this leads to a canonical form which generalizes the two quantum bit schmidt decomposition it is uniquely characterized by the five entanglement parameters it leads to a complete classification of the three quantum bit states it shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties

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

from discrete cucker smale model to continuous cucker smale model in a temperature field we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results we study emergent dynamics of the discrete and continuous cucker smale c s models in a self consistent temperature field and provide refined monocluster flocking estimates more precisely we present a sufficient framework leading to the monocluster flocking toward the constant common temperature and velocity exponentially fast this improves the authors earlier work where existence of common temperature and velocity are not identified our proposed sufficient framework is formulated in terms of the initial data and system parameters using the refined flocking we also provide a uniform in time convergence from the discrete model to the continuous model under some suitable compatibility condition on the asymptotic velocities for the discrete and continuous model as the time step tends to zero we also provide several numerical examples and compare them with analytical results

local symmetry properties of pure three qubit states entanglement types of pure states of three spin particles are classified by means of their stabilizers in the group of local unitary transformations it is shown that the stabilizer is generically discrete and that a larger stabilizer indicates a stationary value for some local invariant we describe all the exceptional states with enlarged stabilizers

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

polynomial invariants of quantum codes the weight enumerators shor and laflamme 1997 of a quantum code are quite powerful tools for exploring its structure as the weight enumerators are quadratic invariants of the code this suggests the consideration of higher degree polynomial invariants we show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one to one correspondence with s sub k sup n we then present a number of equations and inequalities in these invariants in particular we give a higher order generalization of the shadow enumerator of a code and prove that its coefficients are nonnegative we also prove that the quartic invariants of a 4 4 2 sub 2 code are uniquely determined an important step in a proof that any 4 4 2 sub 2 code is additive rains 1999

equilibria and steering laws for planar formations abstract this paper presents a lie group setting for the problem of control of formations as a natural outcome of the analysis of a planar two vehicle formation control law the vehicle trajectories are described using the planar frenet serret equations of motion which capture the evolution of both the vehicle position and orientation for unit speed motion subject to curvature steering control the set of all possible relative equilibria for arbitrary g invariant curvature controls is described where g se 2 is a symmetry group for the control law and a global convergence result for the two vehicle control law is proved an n vehicle generalization of the two vehicle control law is also presented and the corresponding relative equilibria for the n vehicle problem are characterized work is on going to discover stability and convergence results for the n vehicle problem

tensor network methods for invariant theory invariant theory is concerned with functions that do not change under the action of a given group here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states this graphical approach provides an alternative to the polynomial equations that describe invariants which often contain a large number of terms with coefficients raised to high powers this approach also enables one to use known methods from tensor network theory such as the matrix product state mps factorization when studying polynomial invariants as our main example we consider invariants of mpss we generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the r nyi entropies we find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted as well as an alternative and sometimes much simpler means to study polynomial invariants of quantum states in addition many tensor network methods such as mpss contain excellent tools that can be applied in the study of invariants

styled velocity flocking of autonomous vehicles a systematic design this paper develops a measure theoretic approach for generating the collective behavior of flocking of planar autonomous vehicles under all to all communication scheme to obtain flocking protocol we formulate the notion of state of styled velocity flocking for the target state of a design problem instead of considering solution ansatzes this notion captures a broader class of flocking behaviors of non parallel motions as well as embodies the two important performances of cohesion maintenance and collision avoidance to pose a control design problem a measure valued transition equation is obtained to induce an auxiliary one body continuum model from the exact multi body particle model of the collective system the design problem is then formulated on the auxiliary continuum model and a systematic design procedure is presented the state of styled velocity flocking of the particle model under the obtained flocking protocol is then verified with an important result on the continuous dependence on initial data of solutions of the measured valued transition equation application to elliptical flocking of mobile robots is presented to illustrate the novelty of the presented theory

consensus seeking in multi agent systems with multiplicative measurement noises abstract in this paper the consensus problems of the continuous time integrator systems under noisy measurements are considered the measurement noises which appear when agents measure their neighbors states are modeled to be multiplicative by multiplication of the noises here the noise intensities are proportional to the absolute value of the relative states of an agent and its neighbor by using known distributed protocols for integrator agent systems the closed loop system is described in the vector form by a singular stochastic differential equation for the fixed and switching network topology cases constant consensus gains are properly selected such that mean square consensus and strong consensus can be achieved especially exponential mean square convergence of agents states to the common value is derived for the fixed topology case in addition asymptotic unbiased mean square average consensus and asymptotic unbiased strong average consensus are also studied simulations shed light on the effectiveness of the proposed theoretical results

the geometry of multi qubit entanglement this paper a continuation of a previous one iwai 2007 j phys a math theor 40 1361 studies the geometry of multi qubit entanglement with respect to bipartite partitions an n qubit system c2 n is isomorphic with the linear space of 2l 2m complex matrices where l m n according to the isomorphism the local transformation group u 2l u 2m acts on the space of normalized states in let m and g denote the space of normalized states and the local transformation group acting on m respectively according to the orbit types m is stratified into strata among which a principal stratum and the sets of separable states and maximally entangled states will be identified for a function f c det i cc proves to serve as a measure of entanglement where i denotes the 2l 2l identity matrix the f c attains the minimal and the maximal values respectively on the sets of separable states and maximally entangled states and further takes no extremal values on the principal stratum the f c projects to a function on the factor space g m a naturally defined metric on m also projects to that on g m which serves to measure the distance between the separable states f 1 0 and the states f 1 k of prescribed value k of measure to be precise one has to restrict m to the principal stratum when projecting the metric three and four qubit systems will be studied intensively and then multi qubit systems discussed

hilbert kunz functions and frobenius functors we study the asymptotic behavior as a function of e of the lengths of the cohomology of certain complexes these complexes are obtained by applying the e th iterated frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module the rings involved here all have positive prime characteristic for the zeroth homology these functions also contain the class of hilbertkunz functions that a number of other authors have studied this asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper these are defined in terms of the krull dimensions of the matlis duals of the local cohomology of the module there is a more detailed study of this behavior when the given complex is a koszul complex

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

lengths of tors determined by killing powers of ideals in a local ring given a local ring r and n ideals whose sum is primary to the maximal ideal of r one may define a function which takes an n tuple of exponents a1 an to the length of tork r i1 r in these functions are shown to have rational generating functions in certain cases

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

local unitary symmetries and entanglement invariants we investigate the relation between local unitary symmetries and entanglement invariants of multi qubit systems the hilbert space of such systems can be stratified in terms of states with different types of symmetry we review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties on a stratum of a non trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement finding these invariants is often a simpler problem than finding the invariants of the local unitary group the conditions as given by the luna richardson theorem for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed as an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non trivial on the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations this canonical form and the invariants are directly determined by the symmetry group further we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties

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