the ring of conditions of a semisimple group abstract in this paper we are going to give an explicit description of the so called ring of conditions of an adjoint semisimple algebraic group g originally introduced in c de concini c procesi complete symmetric varieties ii intersection theory in algebraic groups and related topics kyoto nagoya 1983 in adv stud pure math vol 6 north holland amsterdam 1985 pp 481 513 in order to study intersection theory in g

tunstall adaptive coding and miscoding in the first part of this paper we tackle the case where a variable length to block tunstall code is used to encode the wrong source miscoding it turns out that exactly as happens in the case with huffman coding the asymptotic excess rate is given by the informational divergence between the probability distribution ruling the source and the probability distribution for which the tunstall code had been devised we also prove asymptotic equality between the individual rate of a codeword and the corresponding self information this allows us to bound the maximal and the minimal length in a tunstall tree in the second part of the paper we study the case where the probability distribution of the source changes in time if this happens it is necessary to frequently update the current code to ensure that the optimality conditions concerning its rate are met this coding procedure is known as adaptive coding we propose some schemes for adaptive tunstall coding based on the structure of the coding tree on the ordering property analogous to gallager s 1978 sibling property of the huffman code and on the tunstall regions analogous to the attraction regions due to longo and galasso 1982

solving a special case of conservative problems by secant like methods we study a class of secant like iterations for solving nonlinear equations in banach spaces a semilocal convergence result is obtained where the first order divided difference of the nonlinear operator is holder continuous for that we use a technique based on a new system of recurrence relations to obtain existence uniqueness domains of the solution and a priori error bounds these results are applied to solve a special case of conservative problems

on the condition number of lagrangian numerical differentiation given distinct nodes u u 0 u 1 u n in 1 1 consider the differentiation rule k 0 h email protected n j 0w email protected hu j where w j 1 k j 0 u 1 j t u p n and 1 j u i u d i j the condition number l k n u s n j 0 w j is a measure of the error due to perturbations in the function values hu j t j rivlin siam j numer anal 1975 theoretically determined sets u which minimize the condition number for fixed k n results of rivlin are used to exhibit an explicit formula for the optimal value of l k n u a description is given of a simple and efficient algorithm and a corresponding fortran subroutine for computing derivatives of interpolating polynomials condition numbers obtained from the subroutine are tabulated for optimal equidistant and geometric nodes and for 1 k n 15 andk n of the same parity the computed values whose relative errors are theoretically bounded by machine precision are remarkably free from roundoff amplification the subroutine can be used as the basis of a simple program that returns approximate derivatives and corresponding error estimates

on the ball of convergence of secant like methods for non differentiable operators in this paper we analyze the local convergence of a uniparametric family of secant like methods for solving nonlinear operators in banach spaces this new study has an important and novel feature since it is applicable to non differential operators so far the results of local convergence usually considered can be only applied to differentiable operators

polytope volume computation a combinatorial form of gram s relation for convex polytopes can be adapted for use in computing polytope volume we present an algorithm for volume computation based on this observation this algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities p x e r ax b as an illustration we compute a formula for the volume of a projective image of the n cube from this formula we deduce that when a and b have rational entries so that the volume of p is also a rational number the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of a and b this settles a question posed by dyer and frieze

extended sufficient semilocal convergence for the secant method we establish new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a banach space using our new concept of recurrent functions and combining lipschitz and center lipschitz conditions on the divided difference operator we obtain a new semilocal convergence analysis of the secant method moreover our sufficient convergence conditions expand the applicability of the secant method in cases not covered before dennis 1971 9 hernandez et al 2005 8 laasonen 1969 15 ortega and rheinboldt 1970 11 potra 1982 5 potra 1985 7 schmidt 1978 18 yamamoto 1987 12 wolfe 1978 19 numerical examples are also provided in this study

on the complexity of computing the volume of a polyhedron we show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix

a general framework for codes involving redundancy minimization a framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function the framework encompasses problems previously proposed by huffman campbell nath and drmota and szpankowski shedding light on the relationships among these problems in particular nath s range of problems can be seen as bridging the minimum average redundancy problem of huffman with the minimum maximum pointwise redundancy problem of drmota and szpankowski using this framework two linear time huffman like algorithms are devised for the minimum maximum pointwise redundancy problem the only one in the framework not previously solved with a huffman like algorithm both algorithms provide solutions common to this problem and a subrange of nath s problems the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and nath problems simple redundancy bounds are also presented

an euler relation for valuations on polytopes abstract a locally finite point set such as the set z n of integral points gives rise to a lattice of polytopes in euclidean space taking vertices from the given point set we develop the combinatorial structure of this polytope lattice and derive euler type relations for valuations on polytopes using the language of mobius inversion in this context a new family of inversion relations is obtained thereby generalizing classical relations of euler dehn sommerville and macdonald

arithmetic multivariate descartes rule let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be any number field or p adic field and consider f f 1 f k where f 1 f k inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i x 1 x n and no more than distinct exponent vectors occur in the monomial term expansions of the f i we prove that f has no more than 1 c n n 3 log n n geometrically isolated roots in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i n where c is an explicit and effectively computable constant depending only on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i this gives a significantly sharper arithmetic analogue of khovanski s theorem on real fewnomials and a higher dimensional generalization of an earlier result of hendrik w lenstra jr for the special case of a single univariate polynomial we also present some further refinements of our new bounds and a higher dimensional generalization of a bound of lipshitz on p adic complex roots connections to non archimedean amoebae and computational complexity including additive complexity and solving for the geometrically isolated rational roots are discussed along the way we thus provide the foundations for an effective arithmetic analogue of fewnomial theory

computing volumes of polyhedra in this note we give two simple methods for calculating the volume of any closed bounded polyhedron in r having an orientable boundary which is triangulated into a set of n 1 dimensional simplices the formulas given require only coordinates of the vertices of the polyhedron 1 introduction the purpose of this note is to give two simple methods for calculating the volume of any closed bounded polyhedron p in r having an orientable boundary ap which is triangulated into a set t of n 1 dimensional simplices following hadwiger 2 we define a polyhedron to be the union of pairwise disjoint convex polyhedra each of which is the convex hull of a finite number of points in 1 we have described an algorithm for obtaining a piecewise linear manifold which closely approximates an implicitly defined manifold if p has been given in such a way then the affine pieces of ap are in general easy to triangulate with an inherited orientation for polyhedra p which are determined by a given system of inequalities methods and programs for triangulating p have been given in 5 6 for such polyhedra a triangulation of the boundary is not easily available so our method is inappropriate of course our approach would also be unnecessary for computing the volume of a parallelotope practical applications of the methods given here may be made to the approxima tion of an area bounded by an implicitly defined curve or to approximation of the volume of a solid which is bounded by an implicitly defined surface although some formulas for the volume of convex polyhedra in rn appear in the literature e g 3 these formulas generally require the computation of the n 1 volume of the facets and additionally they involve extra computations of certain distances the volume formulas we give here involve only the coordinates of the vertices of p

improving the domain of starting points for secant like methods abstract from the simplified secant method and a family of secant like methods we construct a family of predictor corrector iterative methods for solving a nonlinear equation in banach spaces

a polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed we prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d dimensional euclidean space previously such algorithms were known for dimensions d 1 2 3 and 4 only

robust noiseless source coding through a game theoretical approach noiseless coding of a discrete source with partially known statistics is formulated as a two person game the payoff is the average codeword length using shannon codes the code designer picks a source probability distribution for the design of the code while an opponent picks the actual source probability distribution it is shown that if the class of probability distributions allowed is convex then there is a saddle point solution which is determined by the maximum entropy distribution of the convex class the maximum entropy element is derived for three families of source probability mass functions pmf a the class of c contaminated pmf s b the class of pmf s for which each probability is known only through an upper and lower bound c the class of pmf s which is a convex hull of a finite number of known pmf s an extension of the robust noiseless source coding problem for families of sources modeled as first order markov chains is discussed

local convergence of efficient secant type methods for solving nonlinear equations abstract a local convergence analysis for an efficient secant type method stm for solving nonlinear equations is given using both the lipschitz continuous and center lipshchitz continuous divided differences of order one an estimate of the radius of the convergence ball of stm is provided the error estimate matching its convergence order is established numerical examples validating the theoretical results are also provided in the concluding section of this study

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

generalized bc systems based on hermitian vector bundles a generalized bc system associated to a hermitian vector bundle over a riemann surface is introduced in close analogy to the usual rank one case some of the geometric analogies to the well known case are studied in particular if there are no zero modes the non abelian theta divisor appears in the general case where zero modes exist it seems to be more difficult to find a natural description it is discussed in detail why no such system exists in low genus that is on the riemann sphere and on elliptic curves

the incidence algebra of polyhedra over the minkowski algebra abstract the minkowski algebra of polyhedra over an ordered field is introduced and the incidence algebra of polyhedra over this algebra is considered the gram sommerville and gauss bonnet theorems and their analogs for relatively open convex polyhedra are given by various mobius inversion formulas cheeger s chern gauss bonnet density formula for piecewise flat spaces is generalized to non compact and unbounded polyhedra and its non numerical forms are naturally expressed in terms of interior cone chains the tangential densities and interior cone functions are defined as the dual versions of the chern gauss bonnet densities and exterior cone functions the tangential density formulas and their non numerical forms are similarly represented in terms of exterior curvatures and exterior cone chains

computing tropical resultants abstract we fix the supports a a 1 a k of a list of tropical polynomials and define the tropical resultant tr a to be the set of choices of coefficients such that the tropical polynomials have a common solution we prove that tr a is the tropicalization of the algebraic variety of solvable systems and that its dimension can be computed in polynomial time the tropical resultant inherits a fan structure from the secondary fan of the cayley configuration of a and we present algorithms for the traversal of tr a in this structure we also present a new algorithm for recovering a newton polytope from the support of its tropical hypersurface we use this to compute the newton polytope of the sparse resultant polynomial in the case when tr a is of codimension 1 finally we consider the more general setting of specialized tropical resultants and report on experiments with our implementations

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

residue formulae vector partition functions and lattice points in rational polytopes we obtain residue formulae for certain functions of several vari ables as an application we obtain closed formulae for vector partition func tions and for their continuous analogs they imply an euler maclaurin sum mation formula for vector partition functions and for rational convex poly topes as well we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope institut fourier b p 74 38402 saint martin d heres cedex france e mail address mbrion fourier ujf grenoble fr ecole normale superieure 45 rue d ulm 75005 paris cedex 05 france e mail address vergne dmi ens fr

on the semilocal convergence of efficient chebyshev secant type methods we introduce a three step chebyshev secant type method cstm with high efficiency index for solving nonlinear equations in a banach space setting we provide a semilocal convergence analysis for cstm using recurrence relations numerical examples validating our theoretical results are also provided in this study

generalized translation invariant valuations and the polytope algebra abstract we study the space of generalized translation invariant valuations on a finite dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations our main theorem is that mcmullen s polytope algebra is a subalgebra of the partial convolution algebra of generalized translation invariant valuations more precisely we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position the convolution is defined and corresponds to the product in the polytope algebra

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

extending the applicability of secant methods and nondiscrete induction abstract we use nondiscrete mathematical induction to extend the applicability of the secant methods for solving equations in a banach setting our approach has the following advantages over earlier works under the same information weaker sufficient convergence conditions tighter error bounds on the distances involved and more information on the location of the solution numerical examples where our results apply but earlier ones fail to solve nonlinear equation as well as tighter error bounds are also provided in this study

new semilocal and local convergence analysis for the secant method we present a new convergence analysis for the secant method in order to approximate a locally unique solution of a nonlinear equation in a banach space our idea uses lipschitz and center lipschitz instead of just lipschitz conditions in the convergence analysis the new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies such as 2 6 9 11 14 15 17 20 22 26 numerical examples validating the theoretical results are also provided in this study

chebyshev method and convexity from convexity we obtain new convergence theorems for the chebyshev method applied to a nonlinear scalar equation f x 0 moreover kantorovich type conditions for the convergence of the chebyshev method in the complex plane are given finally a new type of assumptions are considered to prove the convergence of this method in banach spaces

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

unified convergence analysis of frozen newton like methods under generalized conditions abstract the objective in this article is to present a unified convergence analysis of frozen newton like methods under generalized lipschitz type conditions for banach space valued operators we also use our new idea of restricted convergence domains where we find a more precise location where the iterates lie leading to at least as tight majorizing functions consequently the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods the conditions do not necessarily imply the differentiability of the operator involved this way our method is suitable for solving equations and systems of equations numerical examples complete the presentation of this article

variable length to variable length source coding a greedy step by step algorithm some results on variable length to variable length source coding are presented a sufficient criterion for the asymptotic optimality of the generic v v code is given this also allows the study of the tunstall huffman scheme performance then a nonoptimal greedy approach to prefix codes is described based on the informational divergence pseudometric and on the gallager algorithm minimization of the current rate associated with the subsequent extensions of the source probability distribution although optimality is not always reached this technique can be usefully employed to improve the tunstall huffman concatenation

local h polynomials invariants of subdivisions and mixed ehrhart theory abstract there are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and ehrhart theory namely the h and h polynomials respectively in this paper we study their generalization to subdivisions and lattice subdivisions of polytopes by abstracting constructions in mixed hodge theory we introduce multivariable polynomials which specialize to the h h polynomials these polynomials the mixed h polynomial and the refined limit mixed h polynomial have rich symmetry non negativity and unimodality properties which both refine known properties of the classical polynomials and reveal new structure for example we prove a lower bound theorem for a related invariant called the local h polynomial we introduce our polynomials by developing a very general formalism for studying subdivisions of eulerian posets that extends the work of stanley brenti and athanasiadis on local h vectors in particular we prove a conjecture of nill and schepers and answer a question of athanasiadis

on the convergence of broyden s method in hilbert spaces abstract in this paper we present a new semilocal convergence analysis for an inverse free broyden s method in a hilbert space setting in the analysis we apply our new idea of recurrent functions concepts of divided differences of order one and lipschitz center lipschitz conditions on the operator involved our analysis extends the applicability of broyden s method in cases not covered before finally we give an example to illustrate the main result in this paper

minimax universal noiseless coding for unifilar and markov sources corresp constructive upper bounds are presented for minimax universal noiseless coding of unifilar sources without any ergodicity assumptions these bounds are obtained by quantizing the estimated probability distribution of source letters with respect to the relative entropy they apply both to fixed length to variable length fv and variable length to fixed length vf codes unifilar sources are a generalization of the usual definition of markov sources so these results apply to markov sources as well these upper bounds agree asymptotically with the lower bounds given by davisson for fv coding of stationary ergodic markov sources

an application of informational divergence to huffman codes a classification of all probability distributions over the finite alphabet of an information source is given where the classes are the sets of distributions sharing the same binary huffman code such a classification can be used in noiseless coding when the distribution of the finite memoryless source varies in time or becomes gradually known instead of applying the huffman algorithm to each new estimate of the probability distribution if a simple test based on the above classification is passed then the huffman code used previously is optimal also for the new distribution

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

target enumeration via euler characteristic integrals we solve the problem of counting the total number of observable targets e g persons vehicles landmarks in a region using local counts performed by a network of sensors each of which measures the number of targets nearby but neither their identities nor any positional information we formulate and solve several such problems based on the types of sensors and mobility of the targets the main contribution of this paper is the adaptation of a topological sheaf integration theory integration with respect to euler characteristic to yield complete solutions to these problems

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

an improved convergence theorem for a class of secant like methods an improved semilocal convergence theorem for a class of secant like methods in banach spaces is provided for solving a nonlinear operator equation it is assumed that the first order divided difference of the nonlinear operator is holder continuous our convergence conditions strategically proposed are weaker than some existing ones and are easily satisfied finally two simple examples are provided to show that our results apply while earlier ones fail

the euler maclaurin formula for simple integral polytopes we give a euler maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope our proof uses elementary methods

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

nonlinear reduction for solving deficient polynomial systems by continuation methods neither continuation methods nor symbolic elimination methods can be directly applied to compute all finite solutions to polynomial systems because the amount of computational time is mostly not proportional to the dimension of the system and to the number of finite solutions the notion of s polynomials is used to developed a reduction algorithm to lower the total degree of the deficient polynomial system so that computing the solutions at infinity can be avoided applying the reduction algorithm before solving the system with continuation methods yields a reliable solution method

valuations on lattice polytopes let be a lattice that is a module of finite rank and let denote the family of convex polytopes with vertices in here convexity refers to the underlying rational vector space in this paper it is shown that any valuation on satisfies the inclusion exclusion principle in the strong sense that appropriate extension properties of the valuation hold indeed the core result is that the class of a lattice polytope in the abstract group for valuations on can be identified with its characteristic function in in fact the same arguments are shown to apply to when is a module of finite rank over an ordered ring and more generally to appropriate families of not necessarily bounded polyhedra

stochastic billiards for sampling from the boundary of a convex set stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the markov chain monte carlo paradigm this paper studies how many steps of the underlying markov chain are required to get samples approximately from the uniform distribution on the boundary of the set for sets with an upper bound on the curvature of the boundary our main theorem implies a polynomial time algorithm for sampling from the boundary of such sets

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

on the possibilistic approach to linear regression models involving uncertain indeterminate or interval data abstract we consider linear regression models where both input data the observations of independent variables and output data the observations of the dependent variable are affected by loss of information caused by uncertainty indeterminacy rounding or censoring instead of real valued crisp data only intervals are available we study a possibilistic generalization of the least squares estimator so called ols set for the interval model investigation of the ols set allows us to quantify whether the replacement of real valued crisp data by interval values can have a significant impact on our knowledge of the value of the ols estimator we show that in the general case very elementary questions about properties of the ols set are computationally intractable assuming p np we also focus on restricted versions of the general interval linear regression model to the crisp input case taking the advantage of the fact that in the crisp input interval output model the ols set is a zonotope we design both exact and approximate methods for its description we also discuss special cases of the regression model e g a model with repeated observations

semilocal convergence of secant like methods for differentiable and nondifferentiable operator equations abstract from well known secant like methods we observe that we can construct a new family of secant like methods that includes the secant method and kurchatov s method we analyse the local orders of convergence and the efficiencies of the methods of the family and study the semilocal convergence for differentiable and nondifferentiable operators finally we apply our results to conservative problems

the k theory of assemblers abstract in this paper we introduce the notion of an assembler which formally encodes cutting and pasting data an assembler has an associated k theory spectrum in which 0 is the free abelian group of objects of the assembler modulo the cutting and pasting relations and in which the higher homotopy groups encode further geometric invariants the goal of this paper is to prove structural theorems about this k theory spectrum including analogs of quillen s localization and devissage theorems we demonstrate the uses of these theorems by analyzing the assembler associated to the grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes

hermitian integral geometry we give in explicit form the principal kinematic formula for the action of the ane unitary group on c n together with a straightforward algebraic method for computing the full array of unitary kinematic formulas expressed in terms of certain convex valuations introduced essentially by h tasaki we introduce also several other canonical bases for the algebra of unitary invariant valuations explore their interrelations and characterize in these terms the cones of positive and monotone elements

an efficient tree decomposition method for permanents and mixed discriminants abstract we present an efficient algorithm to compute permanents mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays tensors we describe the sparsity structure of an array in terms of a graph and we assume that its treewidth denoted as is small our algorithm requires o n 2 arithmetic operations to compute permanents and o n 2 n 3 for mixed discriminants and hyperdeterminants we finally show that mixed volume computation continues to be hard under bounded treewidth assumptions

a convex geometric approach to counting the roots of a polynomial system consider a polynomial system f f1 fn in n variables with complex coefficients a standard way to bound the number of isolated roots of f in cn is to multiply and add the degrees of the fi in a straightfoward computation described explicitly by some variant of bezout s theorem unfortunately even if one knows a priori which monomial terms will appear this upper bound can be far from exact in practice one frequently knows which monomial terms will appear in a polynomial system before one actually decides to count or approximate its roots so a fast accurate root count which takes this additional information into account is vital we propose a much tighter upper bound on the number of isolated roots of f in cn and give an explicit combinatorial geometric criterion for when it is exact as a corollary we obtain a geometric characterization of the average case computational complexity of root counting our root count is a formula derived from the theory of toric varieties and its computation involves finding the mixed volume of n shadowed polytopes in rn our formula generalizes the bkk bound bernshtein 1975 kushnirenko 1976 khovanskii 1978 canny and rojas 1991 in four ways 1 for any r 0 n our formula can be used to count the roots whose first r coordinates are nonzero the bkk bound only handles the r n case as a corollary we obtain that for fixed n all of these root counts for generic f can be done in time polynomial in the numbers of monomial terms of the fi 2 we refine bernshtein s 1975 algebraic criterion for exactness of the bkk bound into a complete classification of the subsets of the coefficients of f whose genericity guarantees exactness in our generalized bound 3 our results are stated in terms of arbitrary algebraically closed fields of any characteristic the bkk bound was originally stated only over the complex numbers 4 we generalize all of the above to results on the number of n k dimensional components of the zero set of f1 fk when k n

on the complexity of computing mixed volumes this paper gives various positive and negative results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension on the negative side we present several p hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes we show that computing the volume of zonotopes is p hard while each corresponding mixed volume can be computed easily but also give examples showing that computing mixed volumes is hard even when computing the volume is easy on the positive side we derive a randomized algorithm for computing the mixed volumes v overbrace k 1 ld k 1 m 1 overbrace k 2 dots k 2 m 2 dots overbrace k s dots k s m s of well presented convex bodies k 1 dots k s where m 1 dots m s in n 0 and m 1 geq n psi n with psi n o frac log n log log n the algorithm is an interpolation method based on polynomial time randomized algorithms for computing the volume of convex bodies this paper concludes with applications of our results to various problems in discrete mathematics combinatorics computational convexity algebraic geometry geometry of numbers and operations research

the gram sommerville and gauss bonnet theorems and combinatorial geometric measures for noncompact polyhedra abstract some non numerical curvature functions are introduced for noncompact and unbounded polyhedcra in an affine space over an ordered field in this paper with these functions the generalized gram sommerville and gauss bonnet theorems are obtained and are found to be dual the euler characteristic can be interpreted as a zero dimensional curvature measure on the space of polyhedra and the higher dimensional geometric measures are similarly obtained the usual numerical curvatures are naturally derived from the non numerical curvature functions

combinatorial mixed valuations abstract combinatorial mixed valuations associated to translation invariant valuations on polytopes are introduced in contrast to the construction of mixed valuations via polarization combinatorial mixed valuations reflect and often inherit properties of inhomogeneous valuations in particular it is shown that under mild assumptions combinatorial mixed valuations are monotone and hence nonnegative for combinatorially positive valuations this has strong computational implications applied to the discrete volume the results generalize and strengthen work of bihan 2015 on discrete mixed volumes for rational polytopes it is proved that combinatorial mixed monotonicity is equivalent to monotonicity stronger even a conjecture is substantiated that combinatorial mixed monotonicity implies the homogeneous monotonicity in the sense of bernig fu 2011

on the average codeword length of optimal binary codes for extended sources corresp although optimal binary source coding using symbol blocks of increasing length must eventually yield a code having an average codeword length arbitrarily close to the source entropy it is known that the sequence of average codeword lengths need not be nonincreasing the sequence is however bounded above by the average codeword length of the source and certain subsequences must be nondecreasing sufficient conditions are obtained describing sources for which a decrease in average codeword length is achieved when coding pairs of symbols a sufficient condition specifying sources for which no such decrease is possible is also obtained

on the local convergence of fast two step newton like methods for solving nonlinear equations a local convergence analysis is presented for a fast two step newton like method tsnlm for solving nonlinear equations in a banach space setting the tsnlm unifies earlier methods such as newton s secant newton like chebyshev secant chebyshev newton steffensen stirling s and other single or multistep methods numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper

on barvinok s algorithm for counting lattice points in fixed dimension we describe a simplification of a recent polynomial time algorithm of a i barvinok for counting the number of lattice points in a polyhedron in fixed dimension in particular we show that only very elementary properties of exponential sums are needed to develop a polynomial time algorithm

h polynomials of zonotopes the ehrhart polynomial of a lattice polytope p encodes information about the number of integer lattice points in positive integral dilates of p the h polynomial of p is the numerator polynomial o

a valuation theoretic approach to translative equidecomposability abstract all simple translation invariant valuations on polytopes are classified as a direct consequence the well known conditions for translative equidecomposability are recovered furthermore a simplified proof of the classification of continuous simple translation invariant valuations is presented

two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex two formulas are exhibited for the multivariate lagrange shape polynomials which interpolate a regular grid on a simplex in r 1 introduction the idea of a shape function or lagrange basis for an approxi mating function subspace is fundamental to the finite element method spline theory and interpolation procedures in general given a set of points or nodes in the domain of the approximants a shape function is an approximant associated with a given node and which assumes the value 1 at that node and the value zero at all other nodes in the set if we have a shape function for each node we say that the set of shape functions is biorthonormal to the node set it is then very natural to attempt to force an approximating function to interpolate the node set i e to take on arbitrary values at the given points by constructing it as a linear combination of the shape functions the coefficients in the linear combination are the arbitrary values to be assumed and the biorthonormal property ensures in an obvious and elemen tary way that this linear combination will indeed interpolate those values at the given nodes less obvious but equally elementary is the fact that biorthonormality of the shape functions implies that they are unique and linearly independent and so form a basis for the approximating subspace to state all this precisely let i tea vector space over an ordered field f of character zero and let f be a family of functions defined on x and with values in f which is itself a vector space over f by means of the operations induced from f let f be an approximating subspace of j of finite dimension tv and let the node set p x x be a subset of x using straightforward arguments of elementary linear algebra as is done by davis in chapter 2 of 1 and thacher 5 6 it is easy to prove the following

finite memory universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence under the constraint that the universal predictor has a finite memory is explored in this analysis the finite memory universal predictors are either deterministic or random time invariant finite state fs machines with k states k state machines the paper provides bounds on the asymptotic achievable regret of these constrained universal predictors as a function of k the number of their states for long enough sequences the specific results are as follows when the universal predictors are deterministic machines the comparison class consists of constant predictors and prediction is with respect to the 0 1 loss function hamming distance we get tight bounds indicating that the optimal asymptotic regret is 1 2k in that case of k state deterministic universal predictors the constant predictors comparison class but prediction is with respect to the self information code length and the square error loss functions we show an upper bound on the regret coding redundancy of o k sup 2 3 and a lower bound of spl theta k sup 4 5 for these loss functions if the predictor is allowed to be a random k state machine i e a machine with random state transitions we get a lower bound of spl theta 1 k on the regret with a matching upper bound of o 1 k for the square error loss and an upper bound of o logk k throughout the paper for the self information loss in addition we provide results for all these loss functions in the case where the comparison class consists of all predictors that are order l markov machines

ehrhart tensor polynomials abstract the notion of ehrhart tensor polynomials a natural generalization of the ehrhart polynomial of a lattice polytope was recently introduced by ludwig and silverstein we initiate a study of their coefficients in the vector and matrix cases we give pick type formulas in terms of triangulations of a lattice polygon as our main tool we introduce h r tensor polynomials extending the notion of the ehrhart h polynomial and for matrices investigate their coefficients for positive semidefiniteness in contrast to the usual h polynomial the coefficients are in general not monotone with respect to inclusion nevertheless we are able to prove positive semidefiniteness in dimension two based on computational results we conjecture positive semidefiniteness of the coefficients in higher dimensions furthermore we generalize hibi s palindromic theorem for reflexive polytopes to h r tensor polynomials and discuss possible future research directions

triangulations of cayley and tutte polytopes cayley polytopes were defined recently as convex hulls of cayley compositions introduced by cayley in 1857 in this paper we resolve braun s conjecture which expresses the volume of cayley polytopes in terms of the number of connected graphs we extend this result to a two variable deformations which we call tutte polytopes the volume of the latter is given via an evaluation of the tutte polynomial of the complete graph our approach is based on an explicit triangulation of the cayley and tutte polytope we prove that simplices in the triangulations correspond to labeled trees and forests the heart of the proof is a direct bijection based on the neighbors first search graph traversal algorithm r esum

on line adaptive canonical prefix coding with bounded compression loss semistatic minimum redundancy prefix mrp coding is fast compared with rival coding methods but requires two passes during encoding its adaptive counterpart dynamic huffman coding requires only one pass over the input message for encoding and decoding and is asymptotically efficient dynamic huffman 1952 coding is however notoriously slow in practice by removing the restriction that the code used for each message symbol must have minimum redundancy and thereby admitting some compression loss it is possible to improve the speed of adaptive mrp coding this paper presents a controlled method for trading compression loss for coding speed by approximating symbol frequencies with a geometric distribution the result is an adaptive mrp coder that is asymptotically efficient and also fast in practice