control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

on existence of limit occupational measures set of a controlled stochastic differential equation we establish that under certain conditions the set of occupational measures as well as the set of mathematical expectations of occupational measures generated by the admissible controls and the corresponding solutions of a controlled stochastic differential equation csde converge with the time horizon tending to infinity to a set called limit occupational measures set loms and we show that this limit set coincides with the set of stationary marginal distributions of the csde we also demonstrate the applicability of our results for averaging of singularly perturbed csde

topological aspects of chow quotients this paper studies the canonical chow quotient of a smooth projective variety by a reductive algebraic group the main purpose is to introduce the perturbation translation specialization relation that gives a computable characterization of the chow cycles of the chow quotient also we provide in the languages that are familiar to topologists and differential geometers many topological interpretations of chow quotient that have the advantage to be more intuitive and geometric more precisely over the field of complex numbers these interpretations are symplectically the moduli spaces of stable orbits with prescribed momentum charges and topologically the moduli space of stable action manifolds

on maximum typical and generic ranks we show that for several notions of rank including tensor rank waring rank and generalized rank with respect to a projective variety the maximum value of rank is at most twice the generic rank we show that over the real numbers the maximum value of the real rank is at most twice the smallest typical rank which is equal to the complex generic rank

limits of nonlinear discrete time control systems with fast subsystems we investigate the asymptotic properties of nonlinear discrete time control systems with fast subsystems in particular we use the averaging method in order to construct a limiting system for the slow subsystem we show continuous dependence of the slow trajectories on the perturbation parameter measuring the ratio of the different step sizes as a byproduct we get an exponential formula for the reachable set of nonlinear continuous time control systems with finite control range

convex functions on symmetric spaces side lengths of polygons and the stability inequalities for weighted configurations at infinity in a symmetric space of noncompact type x g k oriented geodesic segments correspond modulo isometries to vectors in the euclidean weyl chamber we can hence assign vector valued lengths to segments our main result is a system of homogeneous linear inequalities which we call the generalized triangle inequalities or stability inequalities describing the restrictions on the vector valued side lengths of oriented polygons it is based on the mod 2 schubert calculus in the real grassmannians g p for maximal parabolic subgroups p the side lengths of polygons in euclidean buildings are studied in the related paper klm2 applications of the geometric results in both papers to algebraic group theory are given in klm3

induction for secant varieties of segre varieties this paper studies the dimension of secant varieties to segre varieties the problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry an inductive procedure is built around the ideas of successive specializations of points and projections this reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases as applications of the technique we give a complete classification of defective p secant varieties to segre varieties for p 6 we generalize a theorem of catalisano geramita gimigliano on non defectivity of tensor powers of pn we determine the set of p for which unbalanced segre varieties have defective p secant varieties in addition we completely describe the dimensions of the secant varieties to the deficient segre varieties p1 x p1 x pn x pn and p2 x p3 x p3 in the final section we propose a series of conjectures about defective segre varieties

singularly perturbed control systems with one dimensional fast dynamics the order reduction approach to singularly perturbed control systems suggests employing as a variational limit the differential algebraic system obtained when the small parameter is set to be zero it is known that the method is valid only under restrictive convergence conditions on the fast dynamics we verify in this paper that when the fast state variable is one dimensional the order reduction method is valid in general this is true however when appropriate relaxation is allowed in the reduced order system we also indicate how to extract near optimal solutions to the original system from optimal solutions of the order reduction one along the traditional reasoning of separating time scales examples are displayed showing that without allowing the relaxation the order reduction may not provide the correct limit

singular perturbations in linear control systems with weakly coupled stable and unstable fast subsystems abstract a linear control system with slow and fast modes is considered where the different dynamics is represented by a small parameter in the derivatives of the fast states the fast subsystem consists of an asymptotically stable and an unstable part the interaction between which contains a factor proportional to the small parameter in the derivatives the behaviour of the set of trajectories is investigated when the small parameter tends to zero the continuity properties of the optimal value of three classical optimal control problems with control constraints for systems of this type are studied

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

the pauli principle revisited by the pauli exclusion principle no quantum state can be occupied by more than one electron one can state this as a constraint on the one electron density matrix that bounds its eigenvalues by 1 shortly after its discovery the pauli principle was replaced by anti symmetry of the multi electron wave function in this paper we solve a longstanding problem about the impact of this replacement on the one electron density matrix that goes far beyond the original pauli principle our approach uses berenstein and sjamaar s theorem on the restriction of an adjoint orbit onto a subgroup and allows us to treat any type of permutational symmetry

on the validity of the direct sum conjecture the direct sum conjecture states that the multiplicative complexity of disjoint sets of bilinear computations is the sum of their separate multiplicative complexities this conjecture is known to hold for only a few specialized cases in this paper we establish its validity for large classes of computations one such class can be defined as follows let s 1 be a set of r m times n bilinear forms and let s 2 be a different set of s p times q bilinear forms then if 2 in r m n s p q we show that the direct sum conjecture holds over any field the proof involves some nontrivial facts from linear algebra and relies on the theory of invariant polynomials this result also settles the multiplicative complexity of pairs of bilinear forms over any field with large enough cardinality it is also shown that the direct sum conjecture is true for the case when r mn 2

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

differential games with switching strategies abstract a zero sum differential game of infinite horizon is considered positive switching costs are associated with each player we prove that under a condition which is different from the isaacs condition the elliot kalton value of the game always exists the value is the unique viscosity solution of corresponding isaacs type system of equations which appears to be a system of quasi variational inequalities with bilateral obstacles the discussion of the case that the switching costs approach to zero shows that the isaacs condition still has to be assumed to guarantee that the limit of the values corresponding to positive switching costs converge to the value corresponding to zero switching costs the classical elliot kalton value of the game

asymptotic properties of input output operators norm associated with singularly perturbed systems with multiplicative white noise in this paper we study the problem of the asymptotic property of the norm of input output operators related to a class of singularly perturbed stochastic linear systems the system is under perturbation of multiplicative white noise by using reduction order and boundary layer techniques it is shown that the norm of the operator of the perturbed system is less than a given number gamma when the small perturbation varepsilon tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than gamma furthermore a stabilizing robust controller is designed which is independent of perturbation varepsilon

multiscale singularly perturbed control systems limit occupational measures sets and averaging an averaging technique for nonlinear multiscale singularly perturbed control systems is developed issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed general results are illustrated with special cases

on waringʼs problem for systems of skew symmetric forms abstract this paper is devoted to a problem of finding the smallest positive integer s m n k such that m 1 generic skew symmetric k 1 forms in n 1 variables as linear combinations of the same s m n k decomposable skew symmetric k 1 forms this problem is analogous to a well known problem called waring s problem for symmetric forms and can be very naturally translated into a classical problem in algebraic geometry in this paper we will go through some basics of algebraic geometry describe how objects in algebraic geometry can be associated to systems of skew symmetric forms and discuss algebro geometric approaches to establish the existence of triples m n k where s m n k is more than expected

on the ideals of secant varieties to certain rational varieties abstract if x p n is a reduced and irreducible projective variety it is interesting to find the equations describing the higher secant varieties of x in this paper we find those equations in the following cases x p n 1 p n t p n is the segre embedding of the product and n is large with respect to the n i theorem 2 4 the x are some unbalanced segre veronese embeddings x is a del pezzo surface

grassmann secants identifiability and linear systems of tensors abstract for any irreducible non degenerate variety x p r we give a criterion for the k s identifiability of x if k s 1 r then the k s identifiability holds for x if and only if the s identifiability holds for the segre product seg p k x moreover if the s th secant variety of x is not defective and it does not fill the ambient space then we can produce a family of pairs k s for which the k s identifiability holds for x

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

cohomology of regular embeddings we give in this paper an explicit description of the rational cohomology ring of a complete symmetric variety or regular compactification of an algebraic symmetric variety the motivation for this computation comes from the desire to describe an explicit schubert calculus for these varieties which have been considered since the work of chasles and schubert in several special cases the technique we use is the result of our attempts to understand better the work of jurkiewicz and danilov in the case of torus embeddings the combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology the key point is that the reisner stanley algebra of the torus embedding coincides with its equivariant cohomology ring we consider a class of spaces regular embeddings that contains both complete symmetric varieties and smooth torus embeddings by suitably generalizing the notion of reisner stanley algebra we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way

lie algebra representations nilpotent matrices and the c numerical range abstract in this paper we survey some applications of the representation theory of lie algebras to linear algebra this includes the derivation of the jordan form for tensor products of invertible matrices the study of normal form problems for nilpotent matrices as well as the derivation of explicit formulas for the c numerical radius of certain nilpotent block shift matrices arising in quantum mechanics and nuclear magnetic resonance nmr spectroscopy in this latter case a conjecture concerning an explicit formula for the c numerical radius is stated we show the existence of unitary transformations that realize the prospective maxima our approach depends on the clebsch gordan decomposition for unitary representations of su 2 c

on tensors of border rank l in cm n l notice this is the author s version of a work that was accepted for publication in linear algebra and its applications changes resulting from the publishing process such as peer review editing corrections structural formatting and other quality control mechanisms may not be reflected in this document changes may have been made to this work since it was submitted for publication a definitive version was subsequently published in linear algebra and its applications june 23 2011 doi 10 1016 j laa 2011 05 013 the original publication is available at www elsevier com

geometric descriptions of entangled states by auxiliary varieties the aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry in the context of this paper geometric means each stratum of the hilbert space corresponding to an entangled state is an open subset of an algebraic variety built by classical geometric constructions tangent lines secant lines from the set of separable states in this setting we describe well known classifications of multipartite entanglement such as 2 2 n 1 for n 1 quantum systems and a new description with the 2 3 3 quantum system our results complete the approach of miyake and make stronger connections with recent work of algebraic geometers moreover for the quantum systems detailed in this paper we propose an algorithm based on the classical theory of invariants to decide to which subvariety of the hilbert space a given state belongs

on the construction of asymptoticallly optimal controls for singularly perturbed systems we construct asymptotically optimal controls for singularly perturbed nonlinear systems where both the slow and the fast motions are controlled for this purpose we give necessary optimality conditions for an averaged system for the slow motion

minimum variance beamforming with soft response constraints soft constraints on the beamformer response to the signal are examined in the context of minimum variance beamforming a quadratic constraint on the beamformer weights is used to control the mean squared error between a desired response and the actual response in the signal direction the constraint is purposely chosen to permit distortion of the signal with the goal of achieving improved interference cancellation under the assumptions of known signal direction and spectral shape the signal to noise ratio is shown to be a nondecreasing function of the mean square distortion the distortion presented to the signal is easily computed from the beamformer weights and can be equalized after beamforming if desired properties of this beamforming method and its relationship to linear constrained beamforming are discussed simulations verify analytic results and illustrate the utility of the soft constrained approach

on the effect of neglecting sensor dynamics in parameter identification problems in this paper we consider a deterministic parameter identification problem for a nonlinear control system subject to disturbances and with output measured by a sensor with fast dynamics we study the effect of neglecting sensor dynamics on the set of parameter values that are consistent with the measurements we show that when neglecting dynamics we must significantly modify the corresponding static model of the sensor in order to obtain a set of parameter values that contains the true value and is consistent with the measurements we describe a correct model of the static sensor which is in a sense minimal

the carroll and chang conjecture of equal indscal components when candecomp parafac gives perfect fit abstract the candecomp parafac algorithm approximates a set of matrices x 1 x i by products of the form ac i b with c i diagonal i 1 i carroll and chang have conjectured that when the matrices are symmetric the resulting a and b will be column wise proportional for cases of perfect fit ten berge et al have shown that the conjecture holds true in a variety of cases but may fail when there is no unique solution in such cases obtaining proportionality by changing part of the solution seems possible the present paper extends and further clarifies their results in particular where ten berge et al solved all i 2 2 cases now all i 3 3 cases and also the i 4 4 cases for i 2 8 and 9 are clarified in a number of cases a and b necessarily have column wise proportionality when candecomp parafac is run to convergence in other cases proportionality can be obtained by using specific methods no cases were found that seem to resist proportionality

singular perturbations in infinite dimensional control systems in this paper we consider a singularly perturbed control system involving differential inclusions in banach spaces with slow and fast solutions using the averaging approach we obtain sufficient conditions for the hausdorff convergence of the set of slow solutions in the supremum norm we present applications of the theorem to prove convergence of the fast solutions in terms of invariant measures and convergence of equi lipschitz solutions

averaging problem for weakly coupled nonexpansive control systems this paper investigates the limit behavior of a singularly perturbed control system with two state variables which are weakly coupled the main novelty of our averaging approach lies in the fact that the limit dynamic may depend the initial condition of the fast system while in the literature this problem has been usually addressed under conditions that ensure that the limit dynamic is independent to this initial condition our study is based on a suitable nonexpansivity condition on the fast system which generalized dissipativity or stability properties of the fast dynamics

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

maximum principle for a hybrid system via singular perturbations we prove a maximum principle of pontryagin type for the time optimal control of a hybrid system the system is nonlinear and consists of a controlled coupled ode pde the control input acts via the boundary and the interior of the domain systems of this type frequently arise in modeling population dynamics in a contaminated environment for the regularization we use ekeland s variational principle along with a singular perturbation technique for this purpose we introduce a new trajectory on an additional exterior domain whose size may be considered as a singular perturbation parameter it turns out that the maximum principle is stable with respect to the size of the additional exterior domain the technique allows us to obtain necessary optimality conditions without involving measure boundary data

involutions of reductive lie algebras in positive characteristic abstract let g be a reductive group over a field k of characteristic 2 let g lie g let be an involutive automorphism of g and let g k p be the associated symmetric space decomposition for the case of a ground field of characteristic zero the action of the isotropy group g on p is well understood since the well known paper of kostant and rallis b kostant s rallis orbits and representations associated with symmetric spaces amer j math 93 1971 753 809 such a theory in positive characteristic has proved more difficult to develop here we use an approach based on some tools from geometric invariant theory to establish corresponding results in good positive characteristic among other results we prove that the variety n of nilpotent elements of p has a dense open orbit and that the same is true for every fibre of the quotient map p p g however we show that the corresponding statement for g conjectured by richardson is not true we provide a new mostly calculation free proof of the number of irreducible components of n extending a result of sekiguchi for k c finally we apply a theorem of skryabin to describe the infinitesimal invariants k p k

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

a link between quantum entanglement secant varieties and sphericity in this paper we shed light on the relations between three concepts studied in representation theory algebraic geometry and quantum information theory first spherical actions of reductive groups on projective spaces second secant varieties of homogeneous projective varieties and the related notions of rank and border rank third quantum entanglement our main result concerns the relation between the problem of the state reconstruction from its reduced one particle density matrices and the minimal number of separable summands in its decomposition more precisely we show that sphericity implies that states of a given rank cannot be approximated by states of a lower rank we call states for which such an approximation is possible exceptional states for three important from a quantum entanglement perspective cases of distinguishable fermionic and bosonic particles we also show that non sphericity implies the existence of exceptional states remarkably the exceptional states belong to non bipartite entanglement classes in particular we show that the w type states and their appropriate modifications are exceptional states stemming from the second secant variety for three cases above we point out that the existence of the exceptional states is a physical obstruction for deciding the local unitary equivalence of states by means of the one particle reduced density matrices finally for a number of systems of distinguishable particles with a known orbit structure we list all exceptional states and discuss their possible importance in entanglement theory

measurable viability theorems and the hamilton jacobi bellman equation abstract we prove viability and invariance theorems for systems with dynamics depending on time in a measurable way and having time dependent state constraints x t f t x t x t p t in the above t p t is an absolutely continuous set valued map and t x f t x is a set valued map which is measurable with respect to t and upper semicontinuous or continuous or locally lipschitz with respect to x for this aim we investigate infinitesimal generators of reachable maps and the lebesgue points of set valued maps the results are applied to define and to study lower semicontinuous solutions of the hamilton jacobi bellman equation u t h t x u x 0 with the hamiltonian h measurable with respect to time locally lipschitz with respect to x and convex in the last variable

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

differential inclusions with stable subinclusions wherexer te to t andf r x to t g r is a set valued mapping with compact and convex images an absolutely continuous function x m is a solution of 1 if it satisfies 1 for a e t e to t thus a differential inclusion generalizes an ordinary differential equation by permitting a set valued right hand side differential equations containing either control uncertain parameters or discontinuity in the right hand side can be transformed into the framework of 1 l 21 a well known and very useful result due to filippov 3 estimates the uniform distance from an arbitrary absolutely continuous function x to the set of solutions of 1 starting from x t by means of constant times the deviation

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

singular perturbation in mayer s problem for linear systems the well posedness of mayer s problem for a linear system with a small parameter in the derivatives and constrained controls is studied the convergence of the optimal control when the parameter tends to zero is analyzed the state constrained case is discussed

balanced averaging of bilinear systems with applications to stochastic control we study balanced model reduction for stable bilinear systems in the limit of partly vanishing hankel singular values we show that the dynamics admit a splitting into fast and slow subspaces and prove an averaging principle for the slow dynamics we illustrate our method with an example from stochastic control density evolution of a dragged brownian particle and discuss issues of structure preservation and positivity

on comon s conjecture over arbitrary fields abstract comon s conjecture asserts that for a symmetric tensor the rank is equal to the symmetric rank for the field of real numbers r or the field of complex numbers c zhang et al proved the conjecture for the case that the rank of a symmetric tensor does not exceed the order in this paper we show that over arbitrary fields this conjecture is true when the rank of a tensor is less than its order moreover we construct a counterexample in which the rank of a symmetric tensor whose rank equals its order

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

gauged gromov witten theory for small spheres we relate the genus zero gauged gromov witten invariants of a smooth projective variety for sufficiently small area with equivariant gromov witten invariants as an application we deduce a gauged version of abelianization for gromov witten invariants in the small area chamber in the symplectic setting we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map

on distinguished orbits of reductive representations abstract let g be a real reductive lie group and let g gl v be a real reductive representation of g with restricted moment map m g v 0 g in this work we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a g orbit is distinguished i e it contains a critical point of the norm squared of m g we give an elementary proof of the well known convexity theorem of atiyah guillemin sternberg in our particular case and we use it to give an easy to check sufficient condition for a g orbit of an element in a nice space to be distinguished in the case where g is algebraic and is a rational representation the above condition is also necessary making heavy use of recent results of michael jablonski obtaining a generalization of nikolayevsky s nice basis criterion we also provide useful characterizations of nice spaces in terms of the weights of finally some applications to ternary forms are presented

ranks of tensors and a generalization of secant varieties abstract we introduce subspace rank as a tool for studying ranks of tensors and x rank more generally we derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in c 2 c b c b we review the literature from a geometric perspective

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

suboptimal control of singularly perturbed systems and periodic optimization a traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the fast subsystems considered with frozen slow variables and controls it is shown that such an approximation is valid if and only if certain families of periodic optimization problems admit steady state solutions it is also shown how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when approximation by reduced problems is not possible

a unified approach to the design of robust narrow band antenna array processors a unified approach to the design of robust narrowband antenna array processors is presented the approach is based on the idea of minimizing the weighted mean square deviation between the desired response and the response of the processor over variations in parameters three specific examples of robust design are considered robustness against directional mismatch robustness against array geometry errors and robustness against channel phase errors initially a general quadratic constraint on the weights is developed however it is then shown that the quadratic constraint can be replaced by linear constraints or at most linear constraints plus norm constraint these latter constraints are no more complex than those required for designs which do not incorporate robustness features explicitly numerical results show that the proposed approach appears to offer a unified treatment for directly designing narrowband processors which are robust against various types of errors and mismatches between signal model and actual scenario

averaging of three time scale singularly perturbed control systems abstract we investigate the asymptotic properties of singularly perturbed control systems with three time scales we apply the averaging method to construct a limiting system for the slowest motion in the form of a differential inclusion sufficient conditions for the uniform convergence of the slowest trajectories are given

limits of singularly perturbed control problems with statistical dynamics of fast motions we describe the limit behavior of admissible trajectories in a singularly perturbed control system as the small parameter tends to zero a general case is considered where in the limit the fast motion may infinitely rapidly oscillate in time invariant measures of the parameterized fast flow are employed to describe the limit behavior and construct the limit control problem the notion of relatively slow controls is introduced approximating properties of the limit problem within the families of relatively slow controls are verified the results are illustrated by examples

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

properness for scaled gauged maps we prove properness of moduli stacks of gauged maps satisfying a stability conditition introduced by mundet 42 schmitt 48 and ziltener 59 the proof combines a git construction of schmitt 48 properness for twisted stable maps by abramovich vistoli 1 a variation of a boundedness argument due to ciocan fontanine kim maulik 13 and a removal of singularities for bundles on surfaces in colliot thelene sansuc 14

when is a pure state of three qubits determined by its single particle reduced density matrices using techniques from symplectic geometry we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one particle reduced density matrices exactly when their ordered spectra belong to the boundary of the so called kirwan polytope otherwise the states with given reduced density matrices are parameterized up to local unitary equivalence by two real variables given inevitable experimental imprecision this means that already for three qubits a pure quantum state can never be reconstructed from single particle tomography we moreover show that the knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the greenberger horne zeilinger state by stochastic local operations and classical communication and discuss generalizations of our results to an arbitrary number of qubits

rank and optimal computation of generic tensors abstract the typical rank maximal border rank of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated for the size n n 3 with n odd the complement of the set of tensors of maximal border rank is a hypersurface its equation is given

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

the solution to the waring problem for monomials and the sum of coprime monomials abstract in this paper we compute the waring rank of any polynomial of the form f i 1 r m i where the m i are pairwise coprime monomials i e gcd m i m j 1 for i j in particular we determine the waring rank of any monomial as an application we show that certain monomials in three variables give examples of forms of rank higher than the generic form as a further application we produce a sum of power decomposition for any form which is the sum of pairwise coprime monomials

adaptive wideband beamforming with frequency invariance constraints a response variation rv element is introduced to control the consistency of an adaptive wideband beamformer s response over the frequency range of interest by incorporating the rv element into the linearly constrained minimum variance lcmv beamformer we develop a novel linearly constrained beamformer with an improved output signal to interference plus noise ratio sinr compared to both the traditional formulation and the eigenvector based formulation due to an increased number of degrees of freedom for interference suppression in addition two novel wideband beamformers robust against look direction estimation errors are also proposed as a further application of the rv element one is designed by imposing a constraint on the rv element and simultaneously limiting the magnitude response of the beamformer within a pre defined angle range at a reference frequency the other one is obtained by combining the rv element and the worst case performance optimization method both of them are reformulated in a convex form as the second order cone soc programming problem and solved efficiently using interior point method compared with the original robust methods a more efficient and effective control over the beamformer s response at the look direction region is achieved with an improved overall performance

averaging method for discontinuous mayer s problem of singularly perturbed control systems abstract in the present paper we prove that the optimal cost of a mayer s problem for a control system with singular perturbation does converge to the optimal cost of a mayer s problem associated with control system obtained by averaging the main novelty of our result lies on the fact that we did not require any continuity assumption on the final cost

calabi flow geodesic rays and uniqueness of constant scalar curvature kahler metrics we prove that constant scalar curvature kahler metric adjacent to a fixed kahler class is unique up to isomorphism the proof is based on the study of a fourth order evolution equation namely the calabi flow from a new geometric perspective and on the geometry of the space of kahler metrics

limit occupational measures set for a control system and averaging of singularly perturbed control systems abstract we discuss conditions for the existence of the limit occupational measures set for a control system we approximate slow components of the trajectories of a singularly perturbed control system by the solutions of a differential inclusion the differential inclusion is obtained via averaging the slow subsystem over measures from the limit occupational measures set constructed for the associated system describing the fast motions with frozen slow ones

waring loci and the strassen conjecture the waring locus of a form f is the collection of the degree one forms appearing in some minimal sum of powers decomposition of f in this paper we give a complete description of waring loci for several family of forms such as quadrics monomials binary forms and plane cubics we also introduce a waring loci version of strassen s conjecture which implies the original conjecture and we prove it in many cases

on singular perturbations for differential inclusions on the infinite interval abstract we consider a differential inclusion subject to a singular perturbation i e part of the derivatives are multiplied by a small parameter 0 we show that under some stability and structural assumptions every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion obtained for 0 when tends to 0 the goal of the present paper is to provide a new result of tikhonov type on the time interval 0

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

angular momenta of relative equilibrium motions and real moment map geometry chenciner and jimenez perez mosc math j 13 4 621 630 2013 showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general euclidean space form a convex polytope in this note we explain how this result follows from a general convexity theorem of o shea and sjamaar in real moment map geometry math ann 31 415 457 2000 finally we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for riemannian symmetric pairs