lower bounds for parallel algebraic decision trees parallel complexity of convex hulls and related problems we show that any parallel algorithm in the fixed degree algebraic decision tree model that answers membership queries in w r n using p processors requires w n log p n rounds where w is the number of connected components of w we further prove a similar result for the average case complexity we give applications of this result to various fundamental problems in computational geometry like convex hull construction and trapezoidal decomposition and also present algorithms with matching upper bounds

applications of ramsey s theorem to decision tree complexity combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented problems that are defined by simple inequalities between inputs called order invariant problems are considered a decision tree is called k bounded if each query depends on at most k variables no further assumptions on the type of queries are made it is proved that one can replace the queries of any k bounded decision tree that solves an order invariant problem over a large enough input domain with k bounded queries whose outcome depends only on the relative order of the inputs as a consequence all existing lower bounds for comparison based algorithms are valid for general k bounded decision trees where k is a constant an o n log n lower bound for the element uniqueness problem and several other problems for any k bounded decision tree such that k o n c and c n 1 2 bounded decision trees of complexity o n that solve the element uniqueness problem all the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well

oscillation threshold of a clarinet model a numerical continuation approach this paper focuses on the oscillation threshold of single reed instruments several characteristics such as blowing pressure at threshold regime selection and playing frequency are known to change radically when taking into account the reed dynamics and the flow induced by the reed motion previous works have shown interesting tendencies using analytical expressions with simplified models in the present study a more elaborated physical model is considered the influence of several parameters depending on the reed properties the design of the instrument or the control operated by the player are studied previous results on the influence of the reed resonance frequency are confirmed new results concerning the simultaneous influence of two model parameters on oscillation threshold regime selection and playing frequency are presented and discussed the authors use a numerical continuation approach numerical continuation consists in following a given solution of a set of equations when a parameter varies considering the instrument as a dynamical system the oscillation threshold problem is formulated as a path following of hopf bifurcations generalizing the usual approach of the characteristic equation as used in previous works the proposed numerical approach proves to be useful for the study of musical instruments it is complementary to analytical analysis and direct time domain or frequency domain simulations since it allows to derive information that is hardly reachable through simulation without the approximations needed for analytical approach

the g stable pieces of the wonderful compactification let g be a connected simple algebraic group over an algebraically closed field there is a partition of the wonderful compactification of into finite many g stable pieces which was introduced by lusztig in this paper we will investigate the closure of any g stable piece in we will show that the closure is a disjoint union of some g stable pieces which was first conjectured by lusztig we will also prove the existence of cellular decomposition if the closure contains finitely many orbits

the complexity of problems on probabilistic nondeterministic and alternating decision trees this work generalizes decision trees in order to study lower bounds on the running times of algorithms that allow probabilistic nondeterministic or alternating control it is shown that decision trees that are allowed internal randomization at the expense of introducing a small probability of error run no faster asymptotically than ordinary decision trees for a collection of natural problems two geometric techniques from the literature for proving lower bounds on the time required by ordinary decision trees are shown to be special cases of one unified technique that in fact applies to nondeterministic decision trees as well finally it is shown that any lower bound on alternating decision tree time also applies to alternating turing machine time

a purely frequency based floquet hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems abstract since the founding theory established by g floquet more than a hundred years ago computing the stability of periodic solutions has given rise to various numerical methods mostly depending on the way the periodic solutions are themselves determined either in the time domain or in the frequency domain in this paper we address the stability analysis of branches of periodic solutions that are computed by combining a pure harmonic balance method hbm with an asymptotic numerical method anm hbm is a frequency domain method for determining periodic solutions under the form of fourier series and anm is continuation technique that relies on high order taylor series expansion of the solutions branches with respect to a path parameter it is well established now that this hbm anm combination is efficient and reliable provided that the system of ode is first of all recasted with quadratic nonlinearities allowing an easy manipulation of both the taylor and the fourier series in this context hill s method a frequency domain version of floquet theory is revisited so as to become a by product of the hbm applied to a quadratic system allowing the stability analysis to be implemented in an elegant way and with good computing performances the different types of stability changes of periodic solutions are all explored and illustrated through several academic examples including systems that are autonomous or not conservative or not free or forced

parallel comparison merging of many ordered lists we consider the problem of merging m disjoint ordered lists each of size n m we determine up to a constant factor the worst case and average case deterministic and randomized parallel comparison complexity of the problem for all the range of n m and p where p is the number of processors used the worst case deterministic time complexity is log mlog 1 p n loglog nlog 2 p n that means n log mp log log n for p 2n and log mlog p n loglog nlog p n for p 2n clearly merging two equal lists and sorting are special cases of this problem for m 2 and m n respectively we also prove that these bounds hold for randomized algorithms and even for the average case of deterministic or randomized ones therefore the average case of the best deterministic or randomized algorithm for this problem is not faster than the worst case of the best deterministic one by more than a constant factor

elements of a theory of simulation ii sequential dynamical systems we study a class of discrete dynamical systems that is motivated by the generic structure of simulations the systems consist of the following data a a finite graph y with vertex set 1 n where each vertex has a binary state b functions f i f 2 n f 2 n and c an update ordering p the functions f i update the binary state of vertex i as a function of the state of vertex i and its y neighbors and leave the states of all other vertices fixed the update ordering is a permutation of the y vertices by composing the functions f i in the order given by p one obtains the sequential dynamical system sds f y p email protected i 1nf p i f 2 n f 2 n we derive a decomposition result characterize invertible sds and study fixed points in particular we analyse how many different sds that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number finally some specific sds are investigated

an optimal minimum spanning tree algorithm we establish that the algorithmic complexity of the minimumspanning tree problem is equal to its decision tree complexity specifically we present a deterministic algorithm to find aminimum spanning tree of a graph with n vertices andm edges that runs in timeo t m n wheret is the minimum number of edge weightcomparisons needed to determine the solution the algorithm isquite simple and can be implemented on a pointer machine althoughour time bound is optimal the exact function describing it is notknown at present the current best bounds known fort are t m n m and t m n o m m n where is acertain natural inverse of ackermann s function even under theassumption that t is superlinear we show thatif the input graph is selected from gn m our algorithm runs in linear time with high probability regardlessof n m or the permutation of edge weights theanalysis uses a new martingale for gn msimilar to the edge exposure martingale forgn p

verification and sensitivity analysis of minimum spanning trees in linear time komlos has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree the total computational work required by his method is much larger than linear however this paper describes a linear time algorithm for verifying a minimum spanning tree this algorithm combines the result of komlos with a preprocessing and table look up method for small subproblems and with a previously known almost linear time algorithm additionally an optimal deterministic algorithm and a linear time randomized algorithm for sensitivity analysis of minimum spanning trees are presented

the soft heap an approximate priority queue with optimal error rate a simple variant of a priority queue called a soft heap is introduced the data structure supports the usual operations insert delete meld and findmin its novelty is to beat the logarithmic bound on the complexity of a heap in a comparison based model to break this information theoretic barrier the entropy of the data structure is reduced by artifically raising the values of certain keys given any mixed sequence of n operations a soft heap with error rate e for any 0 en of its items have their keys raised the amortized complexity of each operation is constant except for insert which takes 0 log 1 e time the soft heap is optimal for any value of e in a comparison based model the data structure is purely pointer based no arrays are move items across the data structure not individually as is customary but in groups in a data structuring equivalent of car pooling keys must be raised as a result in order to preserve the heap ordering of the data structure the soft heap can be used to compute exact or approximate medians and percentiles optimally it is also useful for approximate sorting and for computing minimum spanning trees of general graphs

flute like musical instruments a toy model investigated through numerical continuation abstract self sustained musical instruments bowed string woodwind and brass instruments can be modelled by nonlinear lumped dynamical systems among these instruments flutes and flue organ pipes present the particularity to be modelled as a delay dynamical system in this paper such a system a toy model of flute like instruments is studied using numerical continuation equilibrium and periodic solutions are explored with respect to the blowing pressure with focus on amplitude and frequency evolutions along the different solution branches as well as jumps between periodic solution branches the influence of a second model parameter namely the inharmonicity on the behaviour of the system is addressed it is shown that harmonicity plays a key role in the presence of hysteresis or quasiperiodic regime throughout the paper experimental results on a real instrument are presented to illustrate various phenomena and allow some qualitative comparisons with numerical results

the effect of number of hamiltonian paths on the complexity of a vertex coloring problem a generalization of dobkin and lipton s element uniqueness problem is introduced for any fixed undirected graph g on vertex set v 1 v 2 cdots v n the problem is to determine given n real numbers x 1 x 2 cdots x n whether x i ne x j for every edge upsilon i upsilon j in g this problem is shown to have upper and lower bounds of theta n log n linear comparisons if g is any dense graph the proof of the lower bound involves showing that any dense graph must contain a subgraph with many hamiltonian paths and demonstrating the relevance of these hamiltonian paths to a geometric argument in addition we exhibit relatively sparse graphs for which the same lower bound holds and relatively dense graphs for which a linear upper bound holds

a minimum spanning tree algorithm with inverse ackermann type complexity a deterministic algorithm for computing a minimum spanning tree of a connected graph is presented its running time is 0 m m n where is the classical functional inverse of ackermann s function and n respectively m is the number of vertices respectively edges the algorithm is comparison based it uses pointers not arrays and it makes no numeric assumptions on the edge costs

a shortest path algorithm for real weighted undirected graphs we present a new scheme for computing shortest paths on real weighted undirected graphs in the fundamental comparison addition model in an efficient preprocessing phase our algorithm creates a linear size structure that facilitates single source shortest path computations in o m log alpha time where alpha alpha m n is the very slowly growing inverse ackermann function m the number of edges and n the number of vertices as special cases our algorithm implies new bounds on both the all pairs and single source shortest paths problems we solve the all pairs problem in o mn log alpha m n time and if the ratio between the maximum and minimum edge lengths is bounded by n log n o 1 we can solve the single source problem in o m n log log n time both these results are theoretical improvements over dijkstra s algorithm which was the previous best for real weighted undirected graphs our algorithm takes the hierarchy based approach invented by thorup

unipotent variety in the group compactification abstract the unipotent variety of a reductive algebraic group g plays an important role in the representation theory in this paper we will consider the closure u of the unipotent variety in the de concini procesi compactification g of a connected simple algebraic group g we will prove that u u is a union of some g stable pieces introduced by lusztig in moscow math j 4 2004 869 896 this was first conjectured by lusztig we will also give an explicit description of u it turns out that similar results hold for the closure of any steinberg fiber in g

the character sheaves on the group compactification abstract we give a definition of character sheaves on the group compactification which is equivalent to lusztig s definition in g lusztig parabolic character sheaves ii mosc math j 4 4 2004 869 896 we also prove some properties of the character sheaves on the group compactification

a randomized time work optimal parallel algorithm for finding a minimum spanning forest we present a randomized algorithm to find a minimum spanning forest msf in an undirected graph with high probability the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write erew pram this result is optimal w r t both work and parallel time and is the first provably optimal parallel algorithm for this problem under both measures we also give a simple general processor allocation scheme for tree like computations

unipotent invariant matrices abstract we describe the variety of fixed points of a unipotent operator acting on the space of matrices we compute the determinant and the rank of a generic symmetric or anti symmetric matrix in the fixed variety yielding information about the generic singular locus of the corresponding bilinear form

response of an artificially blown clarinet to different blowing pressure profiles using an artificial mouth with an accurate pressure control the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally two time profiles are used for the blowing pressure in a first set of experiments the pressure is increased at constant rates then decreased at the same rate in a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time in both cases the experiments are repeated for different increase rates numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure a manifestation of bifurcation delay experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure experimental results also highlight that the speed of the onset transient of the sound is roughly the same independently of the duration of the increase phase of the blowing pressure

explicit mapping of acoustic regimes for wind instruments this paper proposes a methodology to map the various acoustic regimes of wind instruments the maps can be generated in a multidimensional space consisting of design control parameters and initial conditions the boundaries of the maps are obtained explicitly in terms of the parameters using a support vector machine svm classifier as well as a dedicated adaptive sampling scheme the approach is demonstrated on a simplified clarinet model for which several maps are generated based on different criteria examples of computation of the probability of occurrence of a specific acoustic regime are also provided in addition the approach is demonstrated on a design optimization example for optimal intonation

frobenius splitting and geometry of g schubert varieties let x be an equivariant embedding of a connected reductive group g over an algebraically closed field k of positive characteristic let b denote a borel subgroup of g a g schubert variety in x is a subvariety of the form diag g v where v is a b b orbit closure in x in the case where x is the wonderful compactification of a group of adjoint type the g schubert varieties are the closures of lusztig s g stable pieces we prove that x admits a frobenius splitting which is compatible with all g schubert varieties moreover when x is smooth projective and toroidal then any g schubert variety in x admits a stable frobenius splitting along an ample divisors although this indicates that g schubert varieties have nice singularities we present an example of a nonnormal g schubert variety in the wonderful compactification of a group of type g2 finally we also extend the frobenius splitting results to the more general class of schubert varieties

bounding the maximum size of a packet radio network the author addresses the problem arising in the topology design of packet radio networks prns which use time division multiplexing and have a diameter constraint of what is the maximum number n sub c f k of users which can be contained in a diameter k prn with f time slots per frame the author assumed that users cannot transmit and receive simultaneously and cannot transmit receive more than one packet at a time this assumption implies that no two channels accessed by the same user may be assigned the same time slot it is shown that the problem of determining n sub c f k is identical to the problem of determining the largest number of vertices which can be contained in an f edge colorable directed graph with diameter k lower bounds on n sub c f k for f 2 k 3 4 5 are obtained by generating large graphs of this type the graphs are constructed and colored by simple and fast procedures which are similar for different values of f and k an extensive bibliography on the edge coloring problem is included

linear modal stability analysis of bowed strings linearised models are often invoked as a starting point to study complex dynamical systems besides their attractive mathematical simplicity they have a central role for determining the stability properties of static or dynamical states and can often shed light on the influence of the control parameters on the system dynamical behaviour while the bowed string dynamics has been thoroughly studied from a number of points of view mainly by time domain computer simulations this paper proposes to explore its dynamical behaviour adopting a linear framework linearising the friction force near an equilibrium state in steady sliding conditions and using a modal representation of the string dynamics starting from the simplest idealisation of the friction force given by coulomb s law with a velocity dependent friction coefficient the linearised modal equations of the bowed string are presented and the dynamical changes of the system as a function of the bowing parameters are studied using linear stability

intersection cohomology of b b orbit closures in group compactifications an adjoint semi simple group g has a wonderful compactification x which is a smooth projective variety containing g as an open subvariety x is acted upon by g g and b denoting a borel subgroup of g the group b b has finitely many orbits in x the main results of this paper concern the intersection cohomology of the closures of the b b orbits examples of such closures are the large schubert varieties the closures in x of the double cosets bwb in g after recalling some basic results about the wonderful compactification we discuss in section 1 the description of the b b orbits and establish some basic results in section 2 the bruhat order of the set v of orbits is introduced and described explicitly as an application we obtain cellular decompositions of the large schubert varieties let h be the hecke algebra associated to g it is a free module over an algebra of laurent polynomials z u u 1 as a particular case of results of ms the spherical g g variety x defines a representation of the hecke algebra associated to g g i e h z u u 1 h in a free module m over an extension of z u u 1 with a basis mv indexed by v the definition of m is sheaftheoretical working over the algebraic closure of a finite field this is discussed in section 3 on the model of lv a duality map is introduced on m coming from verdier duality in sheaf theory the matrix coefficients of relative to the

on compactifications of the steinberg zero fiber abstract let g be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let x denote an equivariant embedding of g we define a distinguished steinberg fiber n in g called the zero fiber and prove that the closure of n within x is normal and cohen macaulay furthermore when x is smooth we prove that the closure of n is a local complete intersection

soft heaps simplified in 1998 chazelle j acm 47 2000 pp 1012 1027 introduced a new kind of meldable heap priority queue called the soft heap soft heaps trade accuracy for speed the heap operations are allowed to increase the keys of certain items thereby making these items bad as long as the number of bad items in the data structure is at most varepsilon m where m is the total number of insertions performed so far and varepsilon is an error parameter the amortized time per heap operation is o lg frac 1 varepsilon reduced from o lg n where n is the number of items in the heap chazelle used soft heaps in several applications including a faster deterministic minimum spanning tree algorithm and a new deterministic linear time selection algorithm we give a simplified implementation of soft heaps that uses less space and avoids chazelle s dismantling operations we also give a simpler improved analysis that yields an amortized time bound of o lg frac 1 varepsilon for each deleti

character sheaves on the semi stable locus of a group compactification abstract we study the intermediate extension of the character sheaves on an adjoint group to the semi stable locus of its wonderful compactification we show that the intermediate extension can be described by a direct image construction as a consequence we show that the ordinary restriction of a character sheaf on the compactification to a semi stable stratum is a shift of semisimple perverse sheaf and is closely related to lusztig s restriction functor from a character sheaf on a reductive group to a direct sum of character sheaves on a levi subgroup we also provide a conjectural formula for the boundary values inside the semi stable locus of an irreducible character of a finite group of lie type which gives a partial answer to a question of springer 2006 21 this formula holds for steinberg character and characters coming from generic character sheaves in the end we verify lusztig s conjecture lusztig 2004 16 12 6 inside the semi stable locus of the wonderful compactification

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

optimal randomized algorithms for local sorting and set maxima randomized algorithms for two sorting problems are presented in the local sorting problem a graph is given in which each vertex is assigned an element of a total order and the task is to determine the relative order of every pair of adjacent vertices in the set maxima problem a collection of sets whose elements are drawn from a total order is given and the task is to determine the maximum element in each set lower bounds for the problems in the comparison model are described and it is shown that the algorithms are optimal within a constant factor

on the number of upward planar orientations of maximal planar graphs we consider the problem of determining the maximum and the minimum number of upward planar orientations a maximal planar graph can have we show that every n vertex maximal planar graph has at least w 1 189 n and at most o 4 n upward planar orientations moreover we show that there exist n vertex maximal planar graphs having o 2 n upward planar orientations and n vertex maximal planar graphs having w 2 599 n upward planar orientations further we present bounds for the maximum and the minimum number of acyclic orientations a maximal planar graph can have

on symbolic obdd based algorithms for the minimum spanning tree problem the minimum spanning tree problem is one of the most fundamental algorithmic graph problems and obdds are a very common dynamic data structure for boolean functions since in some applications graphs become larger and larger a research branch has emerged which is concerned with the design and analysis of so called symbolic algorithms for classical graph problems on obdd represented graph instances here a symbolic minimum spanning tree algorithm using o log 3 v functional operations is presented where v is the set of vertices of the input graph moreover the computation of the transitive closure is investigated and it is proved that there can be an exponential blow up from input to output size furthermore answering an open problem posed by sawitzki 37 it is shown that every symbolic obdd based algorithm for the minimum spanning tree problem needs exponential space with respect to the obdd size of the input graph this result even holds for planar input graphs

elements of a theory of simulation iii equivalence of sds in the development of mathematical foundations for a theory of simulation a certain class of discrete sequential dynamical systems sds is of particular importance these systems which we refer to as sds consist of a a graph y with vertex set 1 2 n where each vertex has associated a binary state b a vertex labeled set of functions f i y f 2 n f 2 n and c a permutation p s n the function f i y update the state of vertex i as a function of the states of vertex i and its y neighbors and leaves all other states invariant by composing these functions in the order given by p we obtain the sds f y p i 1 nf p i y f 2 n f 2 n in this paper we give a combinatorial upper bound for the number of non equivalent sds for a given graph and we compute this bound explicitly for certain classes of graphs we give a full characterization of invertible sds and analyze the set of fixed points of sequential and parallel cellular automata further we introduce the concept of maj type sds and show that systems of this class only have fixed points as attractors finally we analyze sds that are fixed point free for arbitrary base graphs

elements of a theory of computer simulation i sequential ca over random graphs this paper is a first step in the development of mathematical foundations for a theory of simulation we employ sequentially updated cellular automata sca over arbitrary graphs as a paradigmatic framework in the development of the theory we focus on the properties of causal dependencies among local mappings in a simulation let y be a graph in which a each vertex i has a state x i0 1 and b there exists a local map i defined on the states of the y neighbors and x i that returns the state of y i0 1 suppose i 1 i n is a permutation of the y vertices then the order of application of the local maps i k induces a sequential or asynchronous cellular automaton sca over y in this paper we introduce a mapping between the base graph y over which the sca is defined and which represents the mutual dependencies of the local maps and the update graph u y whose vertices are permutations of all y vertices two permutations p 1 p 2 are adjacent in u y if they differ in exactly two consecutive coordinates i k i k 1 and i k i k 1 ey u y represents a conceptual framework which allows to determine equivalence classes of sca for a given graph x independent of the family of local maps i 1 i n we consider a u as a random variable over g n p and analyze the induced random graph u g n p the update graph and show b that u exhibits properties of a covariant functor

an extension of bruhat s lemma abstract in recent work of lusztig and he g lusztig parabolic character sheaves i mosc math j 4 2004 153 179 g lusztig parabolic character sheaves ii mosc math j 4 2004 869 896 x he unipotent variety in the group compactification adv math 203 2006 109 131 x he the g stable pieces of the wonderful compactification trans amer math soc in press certain decompositions are introduced in the wonderful compactification of an adjoint group g to establish them a combinatorial machinery introduced by bedard is used the present note gives another approach to these results we derive them in section 3 from a result about g an analogue of bruhat s lemma proved in section 2 see theorem 2 6 basic in our approach is the elementary lemma 1 6 the approach can also be used to deal with properties of lusztig s parabolic character sheaves we do not go into this here

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons