polynomial rings over nil rings cannot be homomorphically mapped onto rings with nonzero idempotents abstract we prove that the ring of polynomials in one indeterminate over a nil ring cannot be homomorphically mapped onto a ring containing a nonzero idempotent this result can be regarded as an approximation of a positive solution of kothe s problem

makar limanov s conjecture on free subalgebras abstract it is proved that over every countable field k there is a nil algebra r such that the algebra obtained from r by extending the field k contains noncommutative free subalgebras of arbitrarily high rank it is also shown that over every countable field k there is an algebra r without noncommutative free subalgebras of rank two such that the algebra obtained from r by extending the field k contains a noncommutative free subalgebra of rank two this answers a question of makar limanov lenny makar limanov private communication beijing june 2007

radicals of skew polynomial rings and skew laurent polynomial rings abstract we first introduce the wedderburn radical and the levitzki radical of a ring r where is an automorphism of r using the properties of these radicals we study the wedderburn radical of the skew polynomial ring r x and the skew laurent polynomial ring r x x 1 and next observe the levitzki radical of r x and r x x 1 furthermore we characterize the upper nilradical of r x and r x x 1 via the upper nil radical of r

duo rings and ore extensions abstract we show that in an ore polynomial ring every left ideal is two sided only in the trivial commutative case it is unknown whether the analogous statement about right ideals is true we establish a variety of restrictions on any putative counterexample

motion planning in the presence of moving obstacles this paper investigates the computational complexity of planning the motion of a body b in 2 d or 3 d space so as to avoid collision with moving obstacles of known easily computed trajectories dynamic movement problems are of fundamental importance to robotics but their computational complexity has not previously been investigated we provide evidence that the 3 d dynamic movement problem is intractable even if b has only a constant number of degrees of freedom of movement in particular we prove the problem is pspace hard if b is given a velocity modulus bound on its movements and is np hard even if b has no velocity modulus bound where in both cases b has 6 degrees of freedom to prove these results we use a unique method of simulation of a turing machine that uses time to encode configurations whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom we also investigate a natural class of dynamic problems that we call asteroid avoidance problems b the object we wish to move is a convex polyhedron that is free to move by translation with bounded velocity modulus and the polyhedral obstacles have known translational trajectories but cannot rotate this problem has many applications to robot automobile and aircraft collision avoidance our main positive results are polynomial time algorithms for the 2 d asteroid avoidance problem where b is a moving polygon and we assume a constant number of obstacles as well as single exponential time or polynomial space algorithms for the 3 d asteroid avoidance problem where b is a convex polyhedron and there are arbitrarily many obstacles our techniques for solving these asteroid avoidance problems use normal path arguments which are an intereting generalization of techniques previously used to solve static shortest path problems we also give some additional positive results for various other dynamic movers problems and in particular give polynomial time algorithms for the case in which b has no velocity bounds and the movements of obstacles are algebraic in space time

constructing roadmaps of semi algebraic sets i completeness abstract this paper describes preliminary work on an algorithm for planning collision free motions for a robot manipulator in the presence of obstacles the physical obstacles lead to forbidden regions in the robots configuration space and for collision free motion we need paths through configuration space which avoid these regions our method is to construct a certain one dimensional subset or roadmap of the space of allowable configurations if s denotes the set of allowable configurations the roadmap has the property that any connected component of s contains a single connected component of the roadmap it is also possible starting from an arbitrary point p s to rapidly construct a path from p to a point on the roadmap thus given any two points in s we can rapidly determine whether they lie in the same connected component of s and if they do we can return a candidate path between them we do not give a complete description of the algorithm here but we define the roadmap geometrically and verify that it has the necessary connectivity

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

properties which do not pass to classical rings of quotients abstract we construct examples of ore rings satisfying some standard ring theoretic properties for which the classical rings of quotients do not satisfy those properties examples of properties which do not pass to rings of quotients include abelian dedekind finite semicommutative 2 primal and ni in the process of constructing these examples we correct the literature we also introduce two important construction techniques

an infinite dimensional affine nil algebra with finite gelfand kirillov dimension the famous 1960 s construction of golod and shafarevich yields infinite dimensional nil but not nilpotent algebras however these algebras have exponential growth here we construct an infinite dimensional nil but not locally nilpotent algebra which has polynomially bounded growth

collision detection for moving polyhedra we consider the collision detection problem for a three dimensional solid object moving among polyhedral obstacles the configuration space for this problem is six dimensional and the traditional representation of the space uses three translational parameters and three angles typically euler angles the constraints between the object and obstacles then involve trigonometric functions we show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven dimensional space by simple manipulation the constraints may be projected down into a six dimensional space with no increase in complexity the algebraic form of the constraints greatly simplifies computation of collision points and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles

a vertex algorithm for collision detection abstract an algorithm for detecting the collision of moving objects is presented the algorithm applies to polyhedral objects that can be represented as convex hulls of finite number of vertices in two or three dimensional space this is then extended to a three or four dimensional space respectively to represent the objects and their motion nonlinear programming techniques are then employed to detect possible interference the algorithm detects in one step whether or not the objects will interfere during their motion which may involve pure translations or rotations or both a subalgorithm for computing the minimum distance between the objects is also presented this can be used to solve the interference detection problem or for more general applications

a taxonomy of 2 primal rings abstract various conditions on a noncommutative ring imply that it is 2 primal i e the ring s prime radical coincides with the set of nilpotent elements of the ring we will examine several such conditions and show that their known interdependencies are their only ones of particular interest will be the ps i condition on a ring i e every factor ring modulo the right annihilator of a principal right ideal is 2 primal we will see that even within a fairly narrow class of rings ps i is a strictly stronger condition than 2 primal we will show that the ps i condition is left right asymmetric we will also study the interplay between various types of semilocal rings and various types of 2 primal rings the kothe conjecture will make a cameo appearance in section 6 we will examine subideals of prime ideals of commutative rings that are invariant under derivations

smb collision detection based on temporal coherence the paper presents a novel collision detection algorithm termed the sort moving boxes smb for large number of moving 2d 3d objects which are represented by their axis aligned bounding boxes aabbs the main feature of the algorithm is the full exploitation of the temporal coherence of the objects exhibited in a dynamic environment in the algorithm the aabbs are first projected to each cartesian axis the projected intervals on the axes are separately sorted by the diminishing increment sort dis and further divided into subsections by processing all the intervals within the subsections to check if they overlap a complete contact list can be built the smb is a fast and robust collision detection algorithm particularly for systems involving a large number of moving aabbs and also supports for the dynamic insertion and deletion of objects its performance in terms of both expected total detection time and memory requirements is proportional to the total number of aabbs n and is not influenced by size differences of aabbs the space size and packing density over a large range up to ten times difference the only assumption made is that the sorted list at one time step will remain an almost sorted list at the next time step which is valid for most applications whose movement and deformation of each aabb and the dynamic change of the total number n are approximately continuous

signed frames and hadamard products of gram matrices abstract this paper concerns redundant representations in a hilbert space h of the form f j c j f j j f h these are more general than those obtained from a tight frame and we develop a general theory based on what are called signed frames we are particularly interested in the cases where the scaling factors cj are unique and the geometric interpretation of negative cj this is related to results about the invertibility of certain hadamard products of gram matrices which are of independent interest e g we show for almost every v 1 v n c d rank v i v j r v i v j s min r d 1 d 1 s d 1 d 1 n r s 0 applications include the construction of tight frames of bivariate jacobi polynomials on a triangle which preserve symmetries and numerical results and conjectures about the class of tight signed frames in a finite dimensional space

links between prime ideals in differential operator rings in recent years much work has been done on a localization theory for noetherian rings links between prime ideals play a central role in this development when a ring r is not commutative it is rarely possible to localize r at a prime ideal p having a link between p and another prime ideal is an obstruction to localization in this paper we will investigate the links between prime ideals in differential operator rings showing that it is possible to localize these rings in a certain sense recall that a multiplicatively closed subset s of a ring r is a right ore set if for all r in r and all s in s we have rs n sr qr the set s is an ore set if it is both a right and a left ore set it is always possible to construct a ring rf where the elements of s become units however if we insist on writing the elements of rf as right fractions rs with r in r and s in s then the set s must be a right ore set 9 sect 12 11 let p be a prime ideal of r and let 7 p r e r 1 r p is regular in r p we say that p is a localizable prime ideal if the set u p is an ore set we will shortly give an example of a prime ideal which is not localizable but first we introduce the differential operator ring let r be a ring and let 6 be an additive map from r to itself we call 6 a derivation if it satisfies the product rule 6 ab a b ad b the differential operatqr ring over r denoted r 0 s is a free left r module with basis 1 8 13 thus every element of r 0 s is a polynomial in 0 with coefficients from r the addition in r l s is defined as usual for polynomials but multiplication is extended from r by the rule br rti 6 r the differential operator ring is an associative ring and is determined up to isomorphism by an obvious universal property 9 sect 12 21

asymptotically locally optimal weight vector design for a tighter correlation lower bound of quasi complementary sequence sets a quasi complementary sequence set qcss refers to a set of two dimensional matrices with low nontrivial aperiodic auto and cross correlation sums for multicarrier code division multiple access applications the availability of large qcsss with low correlation sums is desirable the generalized levenshtein bound glb is a lower bound on the maximum aperiodic correlation sum of qcsss the bounding expression of glb is a fractional quadratic function of a weight vector mathbf w and is expressed in terms of three additional parameters associated with qcss the set size k the number of channels m and the sequence length n it is known that a tighter glb compared to the welch bound is possible only if the condition m geq 2 and k geq overline k 1 where overline k is a certain function of m and n is satisfied a challenging research problem is to determine if there exists a weight vector that gives rise to a tighter glb for all not just some k geq overline k 1 and m geq 2 especially for large n i e the condition is asymptotically both necessary and sufficient to achieve this we analytically optimize the glb which is in general nonconvex as the numerator term is an indefinite quadratic function of the weight vector our key idea is to apply the frequency domain decomposition of the circulant matrix in the numerator term to convert the nonconvex problem into a convex one following this optimization approach we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the glb under certain conditions

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

three dimensional sklyanin algebras and grobner bases abstract we consider a sklyanin algebra s with 3 generators which is the quadratic algebra over a field k with 3 generators x y z given by 3 relations p x y q y x r z z 0 p y z q z y r x x 0 and p z x q x z r y y 0 where p q r k this class of algebras enjoyed much of attention in particular using tools from algebraic geometry feigin odesskii 15 and artin tate and van den bergh 3 showed that if at least two of the parameters p q and r are non zero and at least two of three numbers p 3 q 3 and r 3 are distinct then s is koszul and has the same hilbert series as the algebra of commutative polynomials in 3 variables it became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the grobner basis technique the main purpose of this paper is to trace the combinatorial meaning of the properties of sklyanin algebras such as koszulity pbw phs calabi yau and to give a new constructive proof of the above facts due to artin tate and van den bergh further we study a wider class of sklyanin algebras namely the situation when all parameters of relations could be different we call them generalized sklyanin algebras we classify up to isomorphism all generalized sklyanin algebras with the same hilbert series as commutative polynomials on 3 variables we show that generalized sklyanin algebras in general position have a golod shafarevich hilbert series with exception of the case of field with two elements

nilpotent elements and armendariz rings we study the structure of the set of nilpotent elements in armendariz rings and introduce nil armendariz as a generalization we also provide some new examples by proving that if d is a k algebra and n 2 the coproduct d kk x xn 0 is armendariz if and only if d is a domain with k 0 as its group of units finally we study the conditions under which the polynomial ring over a nil armendariz ring is nil armendariz which is related to a question of amitsur

a search algorithm for motion planning with six degrees of freedom abstract the motion planning problem is of central importance to the fields of robotics spatial planning and automated design in robotics we are interested in the automatic synthesis of robot motions given high level specifications of tasks and geometric models of the robot and obstacles the movers problem is to find a continuous collision free path for a moving object through an environment containing obstacles we present an implemented algorithm for the classical formulation of the three dimensional movers problem given an arbitrary rigid polyhedral moving object p with three translational and three rotational degrees of freedom find a continuous collision free path taking p from some initial configuration to a desired goal configuration this paper describes an implementation of a complete algorithm at a given resolution for the full six degree of freedom movers problem the algorithm transforms the six degree of freedom planning problem into a point navigation problem in a six dimensional configuration space called c space the c space obstacles which characterize the physically unachievable configurations are directly represented by six dimensional manifolds whose boundaries are five dimensional c surfaces by characterizing these surfaces and their intersections collision free paths may be found by the closure of three operators which i slide along five dimensional level c surfaces parallel to c space obstacles ii slide along one to four dimensional intersections of level c surfaces and iii jump between six dimensional obstacles these operators are employed by a best first search algorithm in c space we will discuss theoretical properties of the algorithm including completeness at a resolution this paper describes the heuristic search with particular emphasis on the heuristic strategies that evaluate local geometric information at the heart of this paper lie the design and implementation of these strategies for planning paths along level c surfaces and their intersection manifolds and for reasoning about motions with three degrees of rotational freedom the problems of controlling the interaction of these strategies and of integrating diverse local experts for geometric reasoning provide an interesting application of search to a difficult domain with significant practical implications the representations and algorithms we develop impact many geometric planning problems and extend to cartesian manipulators with six degrees of freedom

semi commutativity and the mccoy condition abstract we prove that all reversible rings are mccoy generalizing the fact that both commutative and reduced rings are mccoy we then give an example of a semi commutative ring that is not right mccoy at the same time we also show that semi commutative rings do have a property close to the mccoy condition

efficient motion planning for an l shaped object an algorithm that solves the following motion planning problem is presented given an l shaped body l and a two dimensional region with n point obstacles decide whether there is a continuous motion connecting two given positions and orientations of l during which l avoids collision with the obstacles the algorithm requires o n 2 log 2 n time and o n 2 storage the algorithm is a variant of the cell decomposition technique of the configuration space d leven and m sharir j algorithms 8 1987 pp 192 215 j t schwartz and m sharir comm pure appl math 36 1983 pp 345 398 but it employs a new and efficient technique for obtaining a compact representation of the free space which results in a saving of nearly an order of magnitude the approach used in our algorithm is also applicable to motion planning of certain robotic arms whose spaces of free placements have a structure similar to that of the l shaped body

on the kurosh problem for algebras of polynomial growth over a general field lenagan and smoktunowicz 2007 ls see also lenagan smoktunowicz and young in press lsy gave an example of a nil algebra of finite gelfand kirillov dimension their construction requires a countable base field however we show that for any field k and any monotonically increasing function f n which grows super polynomially but subexponentially there exists an infinite dimensional finitely generated nil k algebra whose growth is asymptotically bounded by f n this construction gives the first examples of nil algebras of subexponential growth over uncountable fields

a new lower bound on the maximum correlation of a set with mismatched filters a new lower bound is proposed in this article like levenshtein bound it relates to the maximum correlation value autocorrelation and cross correlation a set of sequences can achieve the novelty introduced here is that each sequence is associated with a mismatched filter the proposed bound is inspired from levenshtein s holds for any set of unimodular sequences and can be applied in both aperiodic and periodic cases it appears that the obtained expression does not deviate a lot from the matched levenshtein which indicates that the use of a mismatched filter will not guarantee much better sidelobe performance as the number fo sequences is significant contrary to the popular belief

structure and topological conditions of ni rings abstract we first construct an ni ring but not 2 primal from given any 2 primal ring in a simpler way than well known examples we study the structure of ni rings relating to strongly prime ideals and show that minimal strongly prime ideals can be lifted in ni rings a ring is called respectively weakly pm if every respectively strongly prime ideal is contained in a unique maximal ideal in it for a 2 primal ring r sun proved that r is pm if and only if max r is a retract of spec r if and only if spec r is normal in the present note we prove for an ni ring r that r is weakly pm if and only if max r is a retract of sspec r if and only if sspec r is normal where sspec r is the space of strongly prime ideals of r we also prove that r is weakly pm if and only if r is pm when r is a symmetric ring we lastly consider several kinds of extensions of ni rings

fast collision detection in four dimensional space abstract in this paper we consider the collision detection problem for general objects a four dimensional approach is proposed for this problem which detects exactly and in one step when and where the earliest collison will occur between the objects this is done by using four dimensional sets to represent the objects in both space and time the problem is then posed as a nonlinear programming problem the algorithm can handle the case of a rigid body moving on a general path in r 2 or r 3 with simultaneous translation and rotation simulation results on some example problems are given and show that the algorithm is superior to those available in the literature

polynomial rings over nil rings need not be nil abstract we construct a nil algebra over a countable field the polynomial ring over which is not nil this answers a question of amitsur

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

a generic construction of complex codebooks meeting the welch bound codebooks also called signal sets meeting the welch bound on the maximum correlation amplitude are called mwbe codebooks and are desirable in code division multiple access systems two different but related constructions of mwbe codebooks from difference sets were developed by xia and ding recently the objectives of this correspondence are to present a generic construction of mwbe codebooks that contains the previous two constructions as special cases and describe new mwbe codebooks that cannot be produced by the earlier two constructions

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

convex hull representations of models for computing collisions between multiple bodies abstract in this paper we consider a collision detection problem that frequently arises in the field of robotics given a set of bodies with their initial positions and trajectories we wish to identify the first collision that occurs between any two bodies or to determine that none exists for the case of bodies having linear trajectories we construct a convex hull representation of the integer programming model of s z selim and h a almohamad european journal of operational research 119 1 1999 121 129 and compare the relative effectiveness in solving this problem via the resultant linear program we also extend this analysis to model a situation in which bodies move along piecewise linear trajectories possibly rotating at the end of each linear segment for this case we again compare an integer programming approach with its linear programming convex hull representation and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once we provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios as well as the times at which the first collision occurs

generalization and improvement of the levenshtein lower bound for aperiodic correlation this paper deals with lower bounds on aperiodic correlation of sequences it intends to solve two open questions the first one is on the validity of the levenshtein bound for a set of sequences other than binary sequences or those over the roots of unity although this result could be a priori extended to polyphase sequences a formal demonstration is presented here proving that it does actually hold for these sequences the second open question is on the possibility to find a bound tighter than welch s in the case of a set consisting of two sequences m 2 by including the specific structure of correlation sequences a tighter lower bound is introduced for this case besides this method also provides in the cases m 3 and m 4 a tighter bound than the up to now tightest bound provided by liu et al

collision detection of a moving polygon in the presence of polygonal obstacles in the plane this paper presents a new approach for the following collision detection problem in the plane let a simple polygon p rotate at a center spl ogr with constant angular velocity spl omega and translate towards a set of polygonal obstacles s with constant velocity spl nu given p and s as well as their initial positions and given also the velocities of p determine whether or not p will collide with any element of s and report the collided elements of s if collisions occurred an o mn worst case optimal algorithm is proposed to solve this problem where n is the number of vertices of p and m is the number of vertices of the obstacles in s

efficient optimization of the merit factor of long binary sequences autocorrelation sidelobes are a form of self noise that reduce the effectiveness of phase coding in radar and communication systems the merit factor is a well known measure related to the autocorrelation sidelobe energy of a sequence in this paper an equation is derived that allows the change in the merit factor of a binary sequence to be computed for all single element changes with o nlogn operations where n is the sequence length the approach is then extended to multiple element changes allowing the merit factor change to be calculated with an additional o ns3 operations where nsis the number of changed elements irrespective of sequence length the multiple element calculations can be used to update the single element calculations so that in iterative use only o n operations are required per element change to keep the single element calculations current a steep descent algorithm a variation on the steepest descent method employing these two techniques was developed and applied to quarter rotated periodically extended legendre sequences producing optimized sequences with an apparent asymptotic merit factor of approximately 6 3758 modestly higher than the best known prior result of approximately 6 3421 modified jacobi sequences improve after steep descent to an approximate asymptotic merit factor of 6 4382 three prime and four prime jacobi sequences converge very slowly making a determination difficult but appear to have a higher post steep descent asymptotic merit factor than legendre or modified jacobi sequences

a new weight vector for a tighter levenshtein bound on aperiodic correlation the levenshtein bound on aperiodic correlation which is a function of the weight vector is tighter than the welch bound for sequence sets over the complex roots of unity when m 4 and n 2 where m denotes the set size and n the sequence length although it is known that the tightest levenshtein bound is equal to the welch bound for m 1 2 it is unknown whether the levenshtein bound can be tightened for m 3 and levenshtein in his paper published in 1999 postulated that the answer may be negative a new weight vector is proposed in this paper which leads to a tighter levenshtein bound for m 3 n 3 and m 4 n 2 in addition the explicit form of the weight vector which is derived by relating the quadratic minimization to the chebyshev polynomials of the second kind in levenshtein s paper is given interestingly this weight vector also yields a tighter levenshtein bound for m 3 n 3 and m 4 n m a fact not noticed by levenshtein