complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

equidistribution of expanding translates of curves and diophantine approximation on matrices we study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space g gamma of a semisimple algebraic group g we define two families of algebraic subvarieties of the associated partial flag variety g p which give the obstructions to non divergence and equidistribution we apply this to prove that for lebesgue almost every point on an analytic curve in the space of m times n real matrices whose image is not contained in any subvariety coming from these two families dirichlet s theorem on simultaneous diophantine approximation cannot be improved the proof combines geometric invariant theory ratner s theorem on measure rigidity for unipotent flows and linearization technique

a randomized algorithm for closest point queries an algorithm for closest point queries is given the problem is this given a set s of n points in d dimensional space build a data structure so that given an arbitrary query point p a closest point in s to p can be found quickly the measure of distance is the euclidean norm this is sometimes called the post office problem the new data structure will be termed an rpo tree from randomized post office the expected time required to build an rpo tree is o n lceil d 2 rceil 1 epsilon for any fixed epsilon 0 and a query can be answered in o log n worst case time an rpo tree requires o n lceil d 2 rceil 1 epsilon space in the worst case the constant factors in these bounds depend on d and epsilon the bounds are average case due to the randomization employed by the algorithm and hold for any set of input points this result approaches the omega n lceil d 2 rceil worst case time required for any algorithm that constructs the voronoi

asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds extending the earlier results for analytic curve segments in this article we describe the asymptotic behaviour of evolution of a finite segment of a c n smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n manifold in particular we show that if the curve satisfies certain natural geometric conditions the pushforward of the parameter measure on the curve under the ge odesic flow converges to the normalized canonical riemannian measure on the tangent bundle in the limit we also study the limits of geodesic evolution of shrinking segments we use ratner s classification of ergodic invariant measures for unipo tent flows on homogeneous spaces of so n 1 and an observation re lating local growth properties of smooth curves and dynamics of linear sl 2 r actions

symmetry algebras of third order ordinary differential equations the main result of this paper is the complete classification of the third order ordinary differential equations according to their symmetries the same classification was done for second order by tresse determination des invariants ponctuels de l equation differentielle ordinaire du second ordre y x y y gekronte preisschrift hirzel leipzig 1896 and recently for arbitrary order linear ordinary differential equations the sections preceding the classification consist of a brief description of the concepts and methods along the lines of krause and michel lecture notes phys 382 251 1991 these sections also contain some definitions and a table listing the prolongations of a few vector fields finally two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

reduction by invariants and projection of linear representations of lie algebras applied to the construction of nonlinear realizations a procedure for the construction of nonlinear realizations of lie algebras in the context of vessiot guldberg lie algebras of first order systems of ordinary differential equations odes is proposed the method is based on the reduction of invariants and projection of lowest dimensional irreducible representations of lie algebras applications to the description of parameterized first order systems of odes related by contraction of lie algebras are given in particular the kinematical lie algebras in 2 1 and 3 1 dimensions are realized simultaneously as vessiot guldberg lie algebras of parameterized nonlinear systems in r3 and r4 respectively

almost tight upper bounds for vertical decompositions in four dimensions we show that the complexity of the vertical decomposition of an arrangement of n fixed degree algebraic surfaces or surface patches in four dimensions is o n4 e for any e 0 this improves the best previously known upper bound for this problem by a near linear factor and settles a major problem in the theory of arrangements of surfaces open since 1989 the new bound can be extended to higher dimensions yielding the bound o n2d 4 e for any e 0 on the complexity of vertical decompositions in dimensions d 4 we also describe the immediate algorithmic applications of these results which include improved algorithms for point location range searching ray shooting robot motion planning and some geometric optimization problems

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

lie hamilton systems on the plane properties classification and applications we study lie hamilton systems on the plane i e systems of first order differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of planar hamiltonian vector fields with respect to a poisson structure we start with the local classification of finite dimensional real lie algebras of vector fields on the plane obtained in a gonzalez lopez n kamran and p j olver proc london math soc 64 339 1992 and we interpret their results as a local classification of lie systems moreover by determining which of these real lie algebras consist of hamiltonian vector fields with respect to a poisson structure we provide the complete local classification of lie hamilton systems on the plane we present and study through our results new lie hamilton systems of interest which are used to investigate relevant non autonomous differential equations e g we get explicit local diffeomorphisms between such systems in particular the milne pinney second order kummer schwarz complex riccati and buchdahl equations as well as some lotka volterra and nonlinear biomathematical models are analysed from this lie hamilton approach

cycles of linear and semilinear mappings abstract we give a canonical form of matrices of a cycle of linear or semilinear mapping v 1 v 2 v t v 1 in which all v i are complex vector spaces each line is an arrow or and each arrow denotes a linear or semilinear mapping

a point symmetry based method for transforming odes with three dimensional symmetry algebras to their canonical forms we provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three dimensional symmetry algebras which lead to their respective canonical forms

lie hamilton systems on the plane applications and superposition rules a lie hamilton lh system is a nonautonomous system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of hamiltonian vector fields with respect to a poisson structure we provide new algebraic geometric techniques to easily determine the properties of such lie algebras on the plane e g their associated poisson bivectors we study new and known lh systems on with physical biological and mathematical applications new results cover cayley klein riccati equations the here defined planar diffusion riccati systems complex bernoulli differential equations and projective schrodinger equations constants of motion for planar lh systems are explicitly obtained which in turn allow us to derive superposition rules through a coalgebra approach

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

classification of invariant wave equations in this paper we characterize the possible symmetry groups of wave equations and certain evolutionary generalizations in a single time variable and one or more spatial variables furthermore we describe a complete classification of two dimensional wave equations utt f u and potential evolutionary equations uxt f u having a point or contact symmetry group the results rely on lie s classification of planar transformation groups and their relative differential invariants

a new coding based algorithm for finding closest pair of vectors abstract given n vectors x 0 x 1 x n 1 in 0 1 m how to find two vectors whose pairwise hamming distance is minimum this problem is known as the closest pair problem if these vectors are generated uniformly at random except two of them are correlated with pearson correlation coefficient then the problem is called the light bulb problem in this work we propose a novel coding based scheme for the closest pair problem we design both randomized and deterministic algorithms which achieve the best known running time when the length of input vectors m is small and the minimum distance is very small compared to m specifically the running time of our randomized algorithm is o n log 2 n 2 c m poly m and the running time of our deterministic algorithm is o n log n 2 c m poly m where c and c are constants depending only on the relative distance of the closest pair when applied to the light bulb problem our result yields state of the art deterministic running time when the pearson correlation coefficient is very large specifically when 0 9933 our deterministic algorithm runs faster than the previously best deterministic algorithm alman sosa 2019

wildness of the problem of classifying nilpotent lie algebras of vector fields in four variables abstract let f be a field of characteristic zero let w n f be the lie algebra of all f derivations with the lie bracket d 1 d 2 d 1 d 2 d 2 d 1 on the polynomial ring f x 1 x n the problem of classifying finite dimensional subalgebras of w n f was solved if n 2 and f c or f r we prove that this problem is wild if n 4 which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity the structure of finite dimensional subalgebras of w n f is interesting since each derivation in case f r can be considered as a vector field with polynomial coefficients on the manifold r n

a lie systems approach to the riccati hierarchy and partial differential equations abstract it is proved that the members of the riccati hierarchy the so called riccati chain equations can be considered as particular cases of projective riccati equations which greatly simplifies the study of the riccati hierarchy this also allows us to characterize riccati chain equations geometrically in terms of the projective vector fields of a flat riemannian metric and to easily derive their associated superposition rules next we establish necessary and sufficient conditions under which it is possible to map second order riccati chain equations into conformal riccati equations through a local diffeomorphism this fact can be used to determine superposition rules for particular higher order riccati chain equations which depend on fewer particular solutions than in the general case therefore we analyze the properties of euclidean hyperbolic and projective vector fields on the plane in detail finally the use of contact transformations enables us to apply the derived results to the study of certain integrable partial differential equations such as the kaup kupershmidt and sawada kotera equations

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

difference schemes with point symmetries and their numerical tests symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented they have the same three or four dimensional symmetry groups as the original differential equations the new difference schemes are tested as numerical methods the obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost for an example involving a solution with a singularity in the integration region the symmetry preserving scheme contrary to standard ones provides solutions valid beyond the singular point

finding correlations in subquadratic time with applications to learning parities and the closest pair problem given a set of n d dimensional boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have pearson correlation coefficient hamming distance d 1 2 how quickly can one find the two correlated vectors we present an algorithm which for any constant 0 and constant 0 runs in expected time o n5 4 nd applications and extensions of this basic algorithm yield significantly improved algorithms for several other problems approximate closest pair for any sufficiently small constant 0 given n d dimensional vectors there exists an algorithm that returns a pair of vectors whose euclidean or hamming distance differs from that of the closest pair by a factor of at most 1 and runs in time o n2 the best previous algorithms including locality sensitive hashing have runtime o n2 o learning sparse parities with noise given samples from an instance of the learning parities with noise problem where each example has length n the true parity set has size at most k n and the noise rate is there exists an algorithm that identifies the set of k indices in time n 3 k poly 1 1 2 0 4 improves upon the results of grigorescu et al 2011 that give a runtime of n 1 2 2 o 1 k 2 poly 1 1 2 learning k juntas with noise given uniformly random length n boolean vectors together with a label which is some function of just k n of the bits perturbed by noise rate return the set of relevant indices leveraging the reduction of feldman et al 2009 our result for learning k parities implies an algorithm for this problem with runtime n 3 k poly 1 1 2 learning k juntas without noise given uniformly random length n boolean vectors together with a label which is some function of k n of the bits return the set of relevant indices using a modification of the algorithm of mossel et al 2004 and employing our algorithm for learning sparse parities with noise via the reduction of feldman et al 2009 we obtain an algorithm for this problem with runtime n 4 k poly n

comments on superposition rules for nonlinear coupled first order differential equations some comments are made on the classification of finite dimensional subalgebras of the lie algebra of vector fields in n variables and of the related nonlinear ordinary differential equations with superposition principles in particular for n 2 a very natural requirement of indecomposability implies that only two types of equations need be considered

realizations of real low dimensional lie algebras using a new powerful technique based on the notion of megaideal we construct a complete set of inequivalent realizations of real lie algebras of dimension no greater than four in vector fields on a space of an arbitrary finite number of variables our classification amends and essentially generalizes earlier works on the subject

invariant difference schemes and their application to sl 2 hbox mathbb r invariant ordinary differential equations we present an exposition of a method of discretizing ordinary differential equations while preserving their lie point symmetries this method is very general and can be applied to any ordinary differential equations ode with a nontrivial symmetry group the method is applied to obtain numerical solutions of second and third order odes invariant under two different realizations of the symmetry preserving method is shown to provide a better qualitative description of solutions than standard methods in particular it provides solutions that are valid close to singularities and beyond them

on algebraic classification of hermitian quasi exactly solvable matrix schrodinger operators on line we construct six multi parameter families of hermitian quasi exactly solvable matrix schrodinger operators in one variable the method for finding these operators relies heavily upon a special representation of the lie algebra o 2 2 sl 2 sl 2 whose representation space contains an invariant finite dimensional subspace furthermore we select those quasi exactly solvable matrix models that have square integrable eigenfunctions on these models are in direct analogy with the quasi exactly solvable scalar schrodinger operators obtained by turbiner and ushveridze

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

symmetry bounds of variational problems sharp upper bounds on the dimension of the lie algebra of infinitesimal variational and divergence point symmetries of a non trivial lagrangian l x u u u n x u in r of arbitrary order n are found for any given order all lagrangians whose lie algebra of variational or of divergence symmetries is of maximal dimension are completely classified modulo local point transformations it is shown in particular that for n or 2 the algebra of variational symmetries of the generalized free particle lagrangian u n2 is not of maximal dimension whereas when n 1 there are several lagrangians admitting a variational symmetry algebra of maximal dimension and generating differential equations different from the free particle equation a connection between variational problems on the line and scalar evolution equations in one time and one space variables is also established showing that lagrangians with a variational symmetry algebra of maximal dimension correspond to evolution equations with a maximal lie algebra of time preserving time independent infinitesimal point symmetries the technique used in the proof of the above results is applied to give a simple proof of the fact that an ordinary differential equation of order n 2 has a symmetry algebra of maximal dimension if and only if it is locally equivalent under a point transformation to the generalized free particle equation u n 0

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

how to search history this paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times we present a new scheme for recording a history of h updates over an ordered set s of n objects which allows fast neighbor computation at any time in the history the novelty of the method is to allow the set s to be only partially ordered with respect to queries and the time measure to be multi dimensional the generality of the method makes it useful for a number of problems in 3 dimensional geometry for example we are able to give fast algorithms for locating a point in a 3 dimensional complex using linear space or for finding which of n given points is closest to a query plane using a simpler yet conceptually similar technique we show that with o n 2 preprocessing it is possible to determine in o log 2 n time which of n given points in e 3 is closest to an arbitrary query point

a singly exponential stratification scheme for real semi algebraic varieties and its applications abstract this paper describes an effective procedure for stratifying a real semi algebraic set into cells of constant description size the attractive feature of our method is that the number of cells produced is singly exponential in the number of input variables this compares favorably with the doubly exponential size of collins decomposition unlike collins construction however our scheme does not produce a cell complex but only a smooth stratification nevertheless we are able to apply our results in interesting ways to problems of point location and geometric optimization

mixed superposition rules and the riccati hierarchy abstract mixed superposition rules i e functions describing the general solution of a system of first order differential equations in terms of a generic family of particular solutions of first order systems and some constants are studied the main achievement is a generalization of the celebrated lie scheffers theorem characterizing systems admitting a mixed superposition rule this somehow unexpected result says that such systems are exactly lie systems i e they admit a standard superposition rule this provides a new and powerful tool for finding lie systems which is applied here to studying the riccati hierarchy and to retrieving some known results in a more efficient and simpler way

remarks on the classification of a pair of commuting semilinear operators abstract gelfand and ponomarev i m gelfand v a ponomarev remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space funct anal appl 3 1969 325 326 proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k tuples of linear operators for any k we prove an analogous statement for semilinear operators

transdichotomous results in computational geometry i point location in sublogarithmic time given a planar subdivision whose coordinates are integers bounded by u leq2 w we present a linear space data structure that can answer point location queries in o min lg n lg lg n sqrt lg u lg lg u time on the unit cost random access machine ram with word size w this is the first result to beat the standard theta lg n bound for infinite precision models as a consequence we obtain the first o n lg n randomized algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word ram including constructing the convex hull of a three dimensional 3d point set computing the voronoi diagram or the euclidean minimum spanning tree of a planar point set triangulating a polygon with holes and finding intersections among a set of line segments higher dimensional extensions and applications are also discussed though computational geometry with bounded precision input has been investigated for a long time improvements have been limited largely to problems of an orthogonal flavor our results surpass this long standing limitation answering for example a question of willard soda 92

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

the problems of classifying pairs of forms and local algebras with zero cube radical are wild we prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is hermitian pairs of bilinear forms in which the second is symmetric skew symmetric and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n by n matrices up to simultaneous similarity

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

automorphisms of real lie algebras of dimension five or less the lie algebra version of the krull schmidt theorem is formulated and proved this leads to a method for constructing the automorphisms of a direct sum of lie algebras from the automorphisms of its indecomposable components for finite dimensional lie algebras there is a well known algorithm for finding such components so the theorem considerably simplifies the problem of classifying the automorphism groups we illustrate this by classifying the automorphisms of all indecomposable real lie algebras of dimension five or less our results are presented very concisely in tabular form

wildness of the problems of classifying two dimensional spaces of commuting linear operators and certain lie algebras abstract for each two dimensional vector space v of commuting n n matrices over a field f with at least 3 elements we denote by v the vector space of all n 1 n 1 matrices of the form a 0 0 with a v we prove the wildness of the problem of classifying lie algebras v with the bracket operation u v u v v u we also prove the wildness of the problem of classifying two dimensional vector spaces consisting of commuting linear operators on a vector space over a field

optimal search in planar subdivisions a planar subdivision is any partition of the plane into possibly unbounded polygonal regions the subdivision search problem is the following given a subdivision s with n line segments and a query point p determine which region of s contains p we present a practical algorithm for subdivision search that achieves the same optimal worst case complexity bounds as the significantly more complex algorithm of lipton and tarjan namely o log n search time with o n storage our subdivision search structure can be constructed in linear time from the subdivision representation used in many applications

superposition formulas for exterior differential systems abstract in this article we solve an inverse problem in the theory of quotients for differential equations we characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original the fact that a system can be written as a quotient can be used to find the general solution to these equations some examples are given to demonstrate the theory

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

approximate nearest neighbor search for l p spaces 2 p via embeddings abstract while the problem of approximate nearest neighbor search has been well studied for euclidean space and l 1 few non trivial algorithms are known for l p when 2 p in this paper we revisit this fundamental problem and present approximate nearest neighbor search algorithms which give the best known approximation factor guarantees in this setting

simultaneous unitary equivalences abstract let s 1 s 2 s 3 s 4 be given finite sets of pairs of n by n complex matrices we describe an algorithm to determine with finitely many computations whether there is a single unitary matrix u such that each pair of matrices in s 1 is unitarily similar via u each pair of matrices in s 2 is unitarily congruent via u each pair of matrices in s 3 is unitarily similar via u and each pair of matrices in s 4 is unitarily congruent via u

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

a simple algorithm for nearest neighbor search in high dimensions the problem of finding the closest point in high dimensional spaces is common in pattern recognition unfortunately the complexity of most existing search algorithms such as k d tree and r tree grows exponentially with dimension making them impractical for dimensionality above 15 in nearly all applications the closest point is of interest only if it lies within a user specified distance spl epsiv we present a simple and practical algorithm to efficiently search for the nearest neighbor within euclidean distance spl epsiv the use of projection search combined with a novel data structure dramatically improves performance in high dimensions a complexity analysis is presented which helps to automatically determine spl epsiv in structured problems a comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems object recognition is demonstrated as an example application the simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent a c implementation of our algorithm is available

a polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing in this paper we prove several extremal results for geometrically defined hypergraphs in particular we establish an improved lower bound single exponentially decreasing in k on the best constant delta 0 such that the vertex classes p 1 ldots p k of every k partite k uniform semialgebraic hypergraph h p 1 cup cdots cup p k e with e ge varepsilon pi j 1 k p i have for 1 leq i leq k delta p i element subsets p i subseteq p i satisfying p 1 times cdots times p k subseteq e the best previously known lower bound on delta due to bukh and hubard decreased triple exponentially fast in k we give three geometric applications of our results in particular we establish the following strengthening of the so called same type lemma of barany and valtr any disjoint finite sets p 1 ldots p k subset mathbb r d k d have for 1 leq i leq k subsets p i of size at least 2 o d 3k log k p i with the property that every k tuple formed by taking on

on nilpotent and solvable lie algebras of derivations let k be a field and a be a commutative associative k algebra which is an integral domain the lie algebra derka of all k derivations of a is an a module in a natural way and if r is the quotient field of a then rderka is a vector space over r it is proved that if l is a nilpotent subalgebra of rderka of rank k over r i e such that dimrrl k then the derived length of l is at most k and l is finite dimensional over its field of constants in case of solvable lie algebras over a field of characteristic zero their derived length does not exceed 2k nilpotent and solvable lie algebras of rank 1 and 2 over r from the lie algebra rderka are characterized as a consequence we obtain the same estimations for nilpotent and solvable lie algebras of vector fields with polynomial rational or formal coefficients analogously if x is an irreducible affine variety of dimension n over an algebraically closed field k of characteristic zero and ax is its coordinate ring then all nilpotent solvable subalgebras of derkax have derived length at most n 2n respectively

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

k symplectic lie systems theory and applications abstract a lie system is a system of first order ordinary differential equations describing the integral curves of a t dependent vector field taking values in a finite dimensional real lie algebra of vector fields a so called vessiot guldberg lie algebra we suggest the definition of a particular class of lie systems the k symplectic lie systems admitting a vessiot guldberg lie algebra of hamiltonian vector fields with respect to the presymplectic forms of a k symplectic structure we devise new k symplectic geometric methods to study their superposition rules t independent constants of motion and general properties our results are illustrated through examples of physical and mathematical interest as a byproduct we find a new interesting setting of application of the k symplectic geometry systems of first order ordinary differential equations

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian matrix

symmetry preserving discretization of ordinary differential equations large symmetry groups and higher order equations ordinary differential equations odes and ordinary difference systems o ss invariant under the actions of the lie groups and of projective transformations of the independent variables x and dependent variables y are constructed the odes are continuous limits of the o ss or conversely the o ss are invariant discretizations of the odes the invariant o ss are used to calculate numerical solutions of the invariant odes of order up to five the solutions of the invariant numerical schemes are compared to numerical solutions obtained by standard runge kutta methods and to exact solutions when available the invariant method performs at least as well as standard ones and much better in the vicinity of singularities of solutions

wildness for tensors abstract in representation theory a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity the latter problem is considered hopeless it contains the problem of classifying an arbitrary finite system of vector spaces and linear mappings between them we prove that an analogous universal problem in the theory of tensors of order at most 3 over an arbitrary field is the problem of classifying three dimensional arrays up to equivalence transformations a i j k i 1 m j 1 n k 1 t i j k a i j k u i i v j j w k k i 1 m j 1 n k 1 t in which u i i v j j w k k are nonsingular matrices this problem contains the problem of classifying an arbitrary system of tensors of order at most three

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

limiting distributions of curves under geodesic flow on hyperbolic manifolds we consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n manifold under the geodesic flow suppose that the curve is not contained in a stable leaf of the flow it is shown that under the geodesic flow the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold moreover if this immersed submanifold is a proper subset then a lift of the curve to the universal covering space t 1 h n is mapped into a proper subsphere of the ideal boundary sphere h n under the visual map this proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in h n covering the closed immersed submanifold in particular if the visual map does not send a lift of the curve into a proper subsphere of h n then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bun dle of the manifold with respect to the normalized natural rie mannian measure the proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of so n 1

on maximal lie point symmetry groups admitted by scalar stochastic differential equations it is proved that the lie point symmetry group admitted by a scalar stochastic differential equation sde of order n 3 is at most n 2 dimensional this result supplements those for first and second order sdes maximal dimensionality can be used to carry out lie group classifications lie point group classification of the third order sdes is provided

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

geometry of riccati equations over normed division algebras abstract this work introduces and studies riccati equations over finite dimensional normed division algebras we prove that a riccati equation over a finite dimensional normed division algebra a is a particular case of conformal riccati equation on a euclidean space and it can be considered as a curve in a lie algebra of vector fields v so dim a 1 1 previous results on known types of riccati equations are recovered from a new viewpoint a new type of riccati equations the octonionic riccati equations are extended to the octonionic projective line o p 1 as a new physical application quaternionic riccati equations are applied to study quaternionic schrodinger equations on 1 1 dimensions

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators