BIN PACKING WITH CONTROLLABLE ITEM SIZES Abstract. We consider a natural resource allocation problem in which we are given a set of items, where each item has a list of pairs associated with it. Each pair is a configuration of an allowed size for this item, together with a nonnegative penalty, and an item can be packed using any configuration in its list. The goal is to select a configuration for each item so that the number of unit bins needed to pack the sizes plus the sum of penalties is minimized. This problem has applications in operating systems, bandwidth allocation, and discrete time-cost tradeoff planning problems.For the offline version of the problem we design an augmented asymptotic PTAS. That is, an asymptotic approximation scheme that uses bins of size slightly larger than 1. We further consider the online bounded space variant, where only a constant number of bins can be open simultaneously. We design a sequence of algorithms, where the sequence of their competitive ratios tends to the best possible asymptotic competitive ratio. Finally, we study the bin covering problem, in which a bin is covered if the sum of sizes allocated to it is at least 1. In this setting, penalties are interpreted as profits and the goal is to maximize the sum of profits plus the number of covered bins. We design an algorithm of best possible competitive ratio, which is 2. We generalize our results for online algorithms and unit sized bins to the case of variable sized bins, where there may be several distinct sizes of bins available for packing or covering, and get that the competitive ratios are again the same as for the more standard online problems. The classical bin packing problem was introduced in the early 70's [27, 17, 18, 9, 6] . Since then many variants have been studied. In the basic online model, a sequence of items, which are positive numbers no larger than 1, are to be packed into unit sized bins. The sum of items packed into one bin cannot exceed its size and the supply of such bins is unbounded. Each item must be packed into exactly one bin, minimizing the number of bins used. The problem has numerous applications in storage and resource allocation.. Bin packing was investigated in both offline and online scenarios. In the latter scenario, the fact that items arrive online means that each item must be assigned in turn, without knowledge of the next items. We analyze the algorithms using the asymptotic approximation ratio (also called competitive ratio for the case of online algorithms). This is a standard measure for bin packing algorithms. For a given input sequence σ, let A(σ) (or A) be the cost of algorithm A on σ. Let OPT(σ) (or OPT) be the cost of an optimal algorithm, i.e., the minimum possible cost to deal with σ (in the case where online algorithms are studied, this is an offline algorithm which has a complete knowledge of the sequence of items). For minimization problems, the approximation ratio for an algorithm A is defined to be R(A) = lim sup n→∞ sup σ { We have designed an augmented APTAS for bin packing with controllable item sizes. Such an APTAS allows to relax the problem in two ways. The algorithm both uses larger bins (of size 1 + ε) than the ones used by an optimal solution OPT, and the number of bins used is slightly larger (by a factor of 1 + ε plus an additive constant) than the number of bins in OPT.. An open question is whether it possible to use only one such relaxation rather than both of them, that is, whether a dual PTAS exists for the problem, and whether a standard APTAS for the problem can be designed.. Typically, bin packing problems with more than one parameter become hard; this is the case for 2-dimensional vector packing and for 2-dimensional bin packing [30, 2, 5] . On the other hand, some such problems admit an APTAS and an AFPTAS, such as bin packing with rejection [13, 3, 14] and bin packing with cardinality constraints [4, 14] .

B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O


Gerris: A Tree-Based Adaptive Solver For The Incompressible Euler Equations In Complex Geometries An adaptive mesh projection method for the time-dependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volume-of-fluid approach. Second-order convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Planar functions over fields of characteristic two Abstract. Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over Z4. We then specialise to planar monomial functions f (x) = cx t and present constructions and partial results towards their classification. In particular, we show that t = 1 is the only odd exponent for which f (x) = cx t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry. A function f : F q → F q is planar if (1) x. is a permutation of F q for each ǫ ∈ F * q . Planar functions have been introduced by Dembowski and Ostrom [4] to construct finite projective planes and arise in many other contexts. For example, Ganley and Spence [8] showed that planar functions give rise to certain relative difference sets, Nyberg and Knudsen [21] , among others, studied planar functions (under the synonym perfect nonlinear functions) for applications in cryptography, and Carlet, Ding, and Yuan [3] , among others, used planar functions to construct errorcorrecting codes.. Planar functions cannot exist in characteristic two since, if q is even and x is a solution to f (x + ǫ) − f (x) = a for a ∈ F q , then so is x + ǫ. This is the motivation to define a function f : F q → F q to be almost perfect nonlinear if (1) is a 2-to-1 map. Such functions have also been studied extensively for applications in cryptography and coding theory (see Carlet, Charpin, and Zinoviev [2] , for example). However, there is no apparent link between almost perfect nonlinear functions and finite projective planes.. Recently, the second author proposed [23] a concept to overcome the problem that there is no planar function in characteristic two. The definition of a planar function has to be modified as follows. A function f : F 2 n → F 2 n is planar if (2) x → f (x + ǫ) + f (x) + ǫx is a permutation of F 2 n for each ǫ ∈ F * 2 n . Such functions share many of the properties of planar functions in odd characteristic. The next section, which is independent of the rest of this paper, provides further background on planar functions in characteristic two and discusses connections to finite geometries and coding theory.. Recall that every function from F 2 n to itself can be uniquely written as a polynomial function of degree at most 2 n − 2. We consider the simplest nontrivial polynomial functions, namely monomial functions x → cx t for some c ∈ F * 2 n and some integer t. Such functions are often preferred in applications. We are interested in those exponents t satisfying 1 ≤ t ≤ 2 n −2 for which the function x → cx t is planar on F 2 n for some c ∈ F * 2 n . It is readily verified that x → cx 2 k is planar on F 2 n for all c ∈ F * 2 n . We conjecture that the only exponents that give planar monomials are these trivial examples and two further examples, shown in Table 1 . Table 1 . Conjectured complete list of exponents giving planar monomials on F 2 n Exponent t Condition Reference 2 k none trivial 2 k + 1 n = 2k Theorem 6 2 k (2 k + 1) n = 6k Conjecture 7 As in odd characteristic, the complete classification of planar monomials in characteristic two seems to be a challenging problem. This motivates us to study the relaxed problem of classifying monomial functions x → cx t on F 2 n that are planar for at least one c ∈ F * 2 n and infinitely many n. The only known such exponents t are the trivial cases t = 2 k and we conjecture that there are no more. Our main result is the following. Theorem 1. Let t be an odd positive integer. If the function x → cx t is planar on F 2 n for some c ∈ F * 2 n and infinitely many n, then t = 1. The problem of classifying monomial functions x → cx t on F 2 n that are planar for at least one c ∈ F * 2 n and infinitely many n parallels the problem of classifying monomial functions x → x t on F 2 n that are almost perfect nonlinear for infinitely many n. To attack this problem, Janwa, McGuire, and Wilson [13] proposed to use ideas from algebraic geometry. These ideas were further developed by Jedlicka [14] and Hernando and McGuire [10] , leading to a complete solution. The same approach has been used by Hernando and McGuire [11] to prove a conjecture on monomial hyperovals in projective planes and by Leducq [17] and Hernando, McGuire, and Monserrat [12] to give partial results towards a classification of planar monomial functions in odd characteristic (which was recently completed by Zieve [24] ). We shall use a similar approach to prove Theorem 1, though our proof requires several extra ideas.

O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Computing the zeros of quaternion polynomials Abstract-A method is developed to compute the zeros of a quaternion polynomial with all terms of the form ok Xk. This method is based essentially in Niven's algorithm (11, which consists of dividing the polynomial by a characteristic polynomial associated to a zero. The information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial. The companion matrix is represented by a matrix with complex entries. Three numerical examples using Mathematics 2.2 version are given. Concerning the unilateral quaternion polynomials, the problem of counting the number of zeros is well studied by several authors [l-5] . The only reference which gives an explicit way to compute the zeros is Niven's [l] . He arrives at a very simple and neat formula which only depends on two parameters (besides the coefficients of the quaternion polynomial). But Niven's method to obtain these parameters turned, as he said, the algorithm into an unpractical one. Our aim is to show an alternative way to obtain these parameters to apply Niven's formula and make this algorithm practical.. We begin in this section with the definitions of quaternions and quaternion polynomials. In Section 2, we present Niven's algorithm. In Section 3, the definition of similarity is given and some known results concerning quaternion matrices are presented. In Section 4, we define the quaternion companion matrix and present new results which lead, in Section 5, to the proposed tMember of the Centro de Matematica da UBI, Covilha, Portugal. tMember of the I&R.-Polo de Coimbra, Coimbra, Portugal. in the quaternion indeterminate x.. DEFINITION 3. Given a quaternion polynomial p(x), we say that X is a zero if p(X) = 0.. The fundamental theorem of algebra over the algebra of real quaternions was first considered by Eilenberg and Niven [1, 7] . They proved that every quaternion polynomial has at least one zero in W. Later on, Beck [2] proved that if the number of zeros of a polynomial of degree m is finite, then there are exactly m zeros counting possible repetition.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Impact Factor: outdated artefact or stepping-stone to journal certification? A review of Garfield's journal impact factor and its specific implementation as the Thomson Reuters Impact Factor reveals several weaknesses in this commonly-used indicator of journal standing. Key limitations include the mismatch between citing and cited documents, the deceptive display of three decimals that belies the real precision, and the absence of confidence intervals. These are minor issues that are easily amended and should be corrected, but more substantive improvements are needed.There are indications that the scientific community seeks and needs better certification of journal procedures to improve the quality of published science. Comprehensive certification of editorial and review procedures could help ensure adequate procedures to detect duplicate and fraudulent submissions. Increasing scrutiny of the Thomson "systematic, objective means to critically evaluate the world's leading journals, with quantifiable, statistical information based on citation data use of this indicator, despite serious flaws science that attempted to infer human 1955). Although popular in the early measurements offered an inaccurate behaviour. Unlike phrenology, the impact factor (Garfield citation patterns and guiding library purchasing decisions ( However, there are increasing concerns that the impact factor is being used inappropriately and in ways not originally envisaged (Garfield 19 crescendo, as the number of papers that editors celebrate any increase in their index inadequacies of the impact factor bibliometric sciences. Although fit its usefulness. Has it become, like phrenology, a pseudo attempts to draw together key conclusions from many extant papers, to illustrate limitations with explicit examples, and to offer recommendations to It is appropriate to begin by examining exactly h be several variants depending on how this computation is made. Garfield index to denote the generic concept underpinning the impact factor, and the abbreviation TRIF to denote the specific Journal Impact Factor published by Thomson Reuters in their Journal Citation Reports (JCR). The Thomson Reuters impact factor (TRIF) suffers so many weaknesses, that a major overhaul is warranted, and journal editors and other users should cease using the TRIF until Thomson Reuters has addressed these weaknesses. Urgent improvements include the adoption of a 'like-with-like' basis (i.e., citations to articles, divided by the count of articles only), the use of verified one-to-one links only (this would unite authors, editors and Thomson Reuters in quality control); the adoption of a more appropriate reference interval (the present two year interval is too short for many disciplines), the introduction of confidence intervals, and the rounding of reported indices to a more appropriate number of digits. Failing action by Thomson Reuters, journal editors should collaborate as they have come with COPE to introduce a journal certification system that acknowledges procedures to maintain quality: procedures that add value and restrict plagiarism and fraud. The future of quality science communication lies in the hands of editors.

O O O O O O O O B O O O O B I I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B I O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Approximating Center Points with Iterated Radon Points We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in IR d . A point c ∈ IR d is a β-center point of P if every closed halfspace containing c contains at least βn points of P . Every point set has a 1/(d + 1)-center point; our algorithm finds an Ω(1/d2 )-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log 2 d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak -nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly.

O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O B O B O O O O B O O O O B O O O O B O O O O O O B B O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The Push3 execution stack and the evolution of control The Push programming language was developed for use in genetic and evolutionary computation systems, as the representation within which evolving programs are expressed. It has been used in the production of several significant results, including results that were awarded a gold medal in the Human Competitive Results competition at GECCO-2004. One of Push's attractive features in this context is its transparent support for the expression and evolution of modular architectures and complex control structures, achieved through explicit code self-manipulation. The latest version of Push, Push3, enhances this feature by permitting explicit manipulation of an execution stack that contains the expressions that are queued for execution in the interpreter. This paper provides a brief introduction to Push and to execution stack manipulation in Push3. It then presents a series of examples in which Push3 was used with a simple genetic programming system (PushGP) to evolve programs with non-trivial control structures. Complex programs written by humans almost always make use of control structures including conditionals, loops, and recursions. Genetic programming researchers have long appreciated this fact and have developed a number of ways in which programs evolved in genetic programming systems can potentially make use of sophisticated control structures. Some of the earliest work allowed for conditionals ("if then else" structures) and some forms of iterative loops ("do until" structures), and a variety of schemes have been proposed to allow for modules (sometimes also called subroutines, defined functions, automatically defined functions, automatically defined macros, or products of encapsulation); see for example [8, 10, 2, 6, 22, 18, 19] . Additional work has explored the use of implicit or explicit recursion in evolved programs (for example [15, 16, 31, 3, 34, 33, 38, 14, 35, 36, 20, 7, 32, 11] ).. The Push programming language provides an alternative mechanism for the expression and evolution of arbitrary control structures [23, 28, 24, 27] . It is not possible in a paper of this length to compare the Push approach to all of the approaches mentioned in the previous paragraph, and this paper has no such comparative goal; some comparisons can be found in other publications on Push cited above. The objective of this paper is rather to present new features in Push3 that further facilitate the evolution of novel control structures -primarily the facilities for execution stack manipulation -and to demonstrate through a series of examples the kinds of results that the approach can routinely produce. Some of these results are striking both for their power and for their novelty.. The next section of the paper outlines the Push language to provide context for the innovations in Push3 that are described in the subsequent section. The discussion of the Push3 execution stack is followed by a series of case studies in which Push3 was used with a genetic programming system (PushGP) to evolve programs that incorporate non-trivial control structures. As instructions are executed they pop any required input values from the appropriate stacks, preform calculations, and push any output values onto the appropriate stacks. The types of the values that will be needed or produced are specified in the implementations of the instructions themselves, and are independent of the syntactic contexts in which calls to the instructions occur. This scheme ensures that Push instructions will always receive inputs and produce outputs of the appropriate types, regardless of the structure of the programs in which they occur. Whenever an instruction finds insufficient items on the stacks for its inputs it acts as a "no-op" and has no effect.. Instructions in Push3 are typically given names such as <TYPE>.<NAME>, where NAME specifies the operation and TYPE specifies the data type upon which the operation should be preformed. INTEGER.=, for example, takes two input values from the INTEGER stack, compares them, and places the result of the comparison on the BOOLEAN stack. It is not uncommon for the same operator to be implemented for multiple types. The instructions INTEGER.POP, FLOAT.POP, and CODE.POP, for example, each pop the top item from the corresponding stack.. The full Push instruction set is large and cannot be fully documented here, 2 but a sample of some of the more commonly used Push instructions is shown in Table 1 . The instructions shown on the right-hand side are implemented for each of the types described in the left column, so the instruction MAX, for example, exists both as INTEGER.MAX and as FLOAT.MAX.. Flexibility with respect to control arises because CODE is itself a native type in Push. A Push program can put code on the CODE stack (for example, with the CODE.QUOTE instruction), duplicate or otherwise manipulate it, and later execute it by means of other CODE instructions. This allows programs to dynamically create novel control structures and subroutine architectures. Examples of several such results 1 Support for the definition of new types from within Push programs is not part of the current Push3 specification, although several proposals for accomplishing this are under consideration. 2 See [27] . The Push3 EXEC stack supports powerful and parsimonious control regimes through explicit manipulation of the stack of expressions that are queued for execution. These control regimes include standard iteration, several forms of recursion based on code manipulation, combinators, and named subroutines, and less conventional strategies that are difficult to classify. A straightforward genetic programming system that produces Push3 programs (PushGP) can routinely produce solutions that incorporate a range of these control regimes; examples were provided here for reversing and sorting lists and for computing factorials, Fibonacci numbers, powers of 2, and parity. Application of these techniques to real-world problems is currently in progress.. The examples presented here demonstrate the range and novelty of the results that Push3 can routinely produce, but this paper does not provide specific comparisons to other approaches. Comparisons with respect to certain aspects of performance have been provided elsewhere for earlier versions of Push (e.g., scalability and modularization in [28] ); the extension of these studies for Push3 is a topic for future work. The primary contributions of Push3, however, concern the overall range of the results that it can produce and the comparative elegance and generality of the mechanisms by which it can produce them. Comparative assessment of these contributions may be best performed after a larger body of results has been produced.. Implementations of the Push programming language and the PushGP genetic programming system are available freely from the Push project home page.

O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O


Braess's Paradox for Flows Over Time Abstract. We study the properties of Braess's paradox in the context of the model of congestion games with flow over time introduced by Koch and Skutella. We compare them to the well known properties of Braess's paradox for Wardrop's model of games with static flows. We show that there are networks which do not admit Braess's paradox in Wardrop's model, but which admit it in the model with flow over time. Moreover, there is a topology that admits a much more severe Braess's ratio for this model. Further, despite its symmetry for games with static flow, we show that Braess's paradox is not symmetric for flows over time. We illustrate that there are network topologies which exhibit Braess's paradox, but for which the transpose does not. Finally, we conjecture a necessary and sufficient condition of existence of Braess's paradox in a network, and prove the condition of existence of the paradox either in the network or in its transpose. Selfish routing and congestion games on networks have been analyzed mainly with respect to only static flows. The most prevalent model of congestion games with static flows is Wardrop's model [7, 18] extensively studied by Roughgarden and Tardos [15, 16] . A game in Wardrop's model is played by an infinite set of players each of which is selfishly routing only a negligible amount of his traffic.. In various applications, e.g. road traffic control and communication networks, however, flow variation over time is a crucial feature. Flow congestion on links and the time to traverse them may change over time in such applications and the flow does not reach its destination instantaneously, but it travels through the network at a certain speed determined by link transit times. We model these phenomena by flows over time (also known as dynamic flows) introduced by Ford and Fulkerson [5, 6] . Nash equilibria for flows over time were introduced by Vickrey [17] and Yagar [19] and mainly studied within the traffic community. For a survey see, e.g. [13] . In 2009, Koch and Skutella [9] , defined a new variant of flows over time and introduced the notion of the price of anarchy for them. This model is based on the deterministic queueing model introduced by Vickrey [17] , in which if at some point in time, more flow tries to enter a link than its capacity allows, the flow queues up at the link tail and waits until it may actually enter the link. The total time spent by a flow particle to traverse a single link is then the sum of the waiting time in the link queue and the actual time to traverse the link. In the model with flow over time introduced by Koch and Skutella, every link of a given network has a fixed capacity and a fixed free flow transit time. The link capacity bounds the maximal rate at which the flow may traverse the link and the link free flow transit time expresses the time a flow particle spends traveling from the link tail to its head.. It is a well known property of selfish routing with static flows that adding a new link to a network does not necessarily decrease the congestion in Nash equilibrium, but, paradoxically, it may even increase it, and so increase the cost of routing through the network. This phenomenon, discovered by Braess [2] , is called Braess's paradox. For a survey see, e.g. [14] .. For Wardrop's model of static flows, it is known [10, 14] that the ratio by which the efficiency of a network may improve by removing any number of its links, i.e. Braess's ratio, is at most ⌊n/2⌋, where n is the number of network nodes. This bound is tight [8, 14] for Wardrop's model. In this paper, we will prove that this bound does not generalize for the model of games with flow over time, and that there is a topology which admits a much more severe Braess's ratio. For flows over time nothing was known in this respect. Akamatsu and Heydecker [1] considered similar paradoxes for flows over time from a different point of view.. In principle, the only kind of topology that admits Braess's paradox in static flows is the Wheatstone network, see Fig. 1(a) , also known as the θ-network. For Wardrop's model of static flows, it is known [12] that Braess's paradox may arise on and only on networks which contain the Wheatstone network as a topological minor. Recall, that a network H is called a topological minor of a network G if a subdivision of H is isomorphic to a subgraph of G. The networks that do not contain the Wheatstone network as a topological minor are usually called series-parallel [4] , as they can be inductively composed by a number of series and parallel compositions from an edge. In other words, a network admits Braess's paradox in Wardrop's model if and only if it is not series-parallel. We will show that there are networks which do not admit Braess's paradox in Wardrop's model, but which admit it in the model with flow over time. Thus, we will introduce a class of series-parallel networks, for which there are instances of the model with flow over time that admit Braess's paradox. An example of such a network is shown in Fig. 1(b) . Actually, all these networks are even extension-parallel. As the Wheatstone network is symmetric, or more precisely, as it is isomorphic to its transpose, a network G admits Braess's paradox in Wardrop's model if and only if its transpose G T admits Braess's paradox as well. Moreover, we know that an instance of Wardrop's model on a network G admits Braess's paradox if and only if the instance on the transpose network G T with the same latency functions and the same flow supply admits it as well.. We will illustrate that in the model with flow over time there exist network topologies which exhibit Braess's paradox, but for which the transpose does not. Also, we will show that there is an infinite set of instances of the model with flow over time, which admit Braess's paradox, but none of the corresponding instances on their transpose networks with the same traffic supply and capacities and the same free flow transit times admits it any more. At the end of this paper, we conjecture a necessary and sufficient condition of existence of Braess's paradox in a network, and prove the condition of existence of the paradox either in the network or in its transpose.. Further discussion on the results can also be found in Macko's PhD thesis [11] .. The structure of this paper is as follows: In Section 2, we provide the formal definition of Koch's and Skutella's model of games with flow over time and define all notations we use later in this paper. Then, in Section 3, we prove the lower bound on Braess's ratio for the model of games with flow over time, and we show that there are networks which admit Braess's paradox in this model, but which do not admit it in Wardrop's model. In Section 4, we show that Braess's paradox is not symmetric in this model. And finally, in Section 5, we provide the necessary and sufficient conditions of existence of Braess's paradox in a network. We have proved several new properties of Braess's paradox for congestion games with flow over time. However, a number of questions have been left open.. We showed that there are networks which do not admit Braess's paradox in games with static flows, but which admit it in the model with flow over time. We showed that these networks admit a much more severe Braess's ratio for this model. In particular, we showed that Braess's ratio of the class of all instances of games with flow over time on networks with n nodes is at least n − 1. What is the upper bound on Braess's ratio for this model?. Then, we illustrated that Braess's paradox is not symmetric for flows over time, although it is symmetric for the case of static flows. We showed that there are network topologies which exhibit Braess's paradox, but for which the transpose does not. Is this asymmetry of Braess's paradox inherent for flows over time? What are the properties of Braess's paradox for different models of games with flows over time?. Finally, we conjectured a necessary and sufficient condition of existence of Braess's paradox in a network, and proved the condition of existence of the paradox either in the network or in its transpose. Is this conjecture valid in general?

B I O B I I O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O B I I O O O O O B I O O O O O O B I I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O


A simulation study of xcp-b performance in wireless multi-hop networks XCP-b proposes a modification to the XCP router algorithm that computes the spare bandwidth. The modification removes the need for an XCP router to know the exact capacity of the channel, making it possible to use the XCP-b variant in transmission media where the capacity is hard to measure. An example of this kind of medium is the IEEE 802.11. Previous work shows that XCP-b behaves well in single-hop wireless networks and that it outperforms TCP in terms of fairness, queuing delay, stability and efficiency when the bandwidth delay product of the network grows. In this paper we extend the validation and evaluation of XCP-b to the case of multi-hop wireless networks, both stand-alone and as access networks to other wired networks.The results show that XCP-b maintains its fundamental characteristics in wireless multi-hop scenarios, such as stable throughput and low standing queues, while distributing the bandwidth fairly and using the available capacity efficiently. The simulations also show that XCP-b produces congestion window values that are closer than TCP to the theoretical upper-bound which maximizes spatial reuse. Van Jacobson's algorithm [10] used by TCP [16] has been the main responsible for congestion control in the Internet Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. since the early 90's. Although it has performed its task remarkably well, it has some characteristics that degrade the QoS of the network. Such characteristics include increased queuing delay, unstable throughput, limited fairness and slow and inefficient behaviour in high bandwidth delay product networks. The root of the problem is the fact that TCP's algorithm relies on rare events which carry low resolution information. This leaves too much guessing to be done by TCP, resulting in TCP making inaccurate decisions.. XCP [12] proposes a new approach, where routers take an active role in informing sources about the state of the network, and how to adjust the sending rate, giving precise feedback. XCP flows have a stable throughput and achieve fair bandwidth allocation, the bottleneck capacity is fully utilized for any bandwidth delay product network, and the bottleneck queue stabilizes at zero length. In order to calculate the precise feedback for each source, the XCP controller needs to compute the current spare bandwidth of the link, and for that the XCP controller needs to be configured with the capacity of the outgoing link. In shared-access multirate media, knowing the exact capacity of the link and the fair share of each station is difficult, thus XCP struggles to work in such media.. XCP-b is an alternative to XCP, which is better suited for these media where the capacity is not easy to measure. It proposes that the spare capacity of the link is computed by measuring variations of the persistent queue of the outgoing link. The XCP-b algorithm has been validated in singlehop wireless scenarios and showed that XCP properties are maintained: stable throughput, low queuing, accurate fairness, and high efficiency as the bandwidth delay product scales.. In this paper we extend the evaluation of XCP-b to scenarios including wireless multi-hop scenarios both functioning as access networks or stand-alone networks. The paper is organized as follows. § 3 describes briefly the operation of XCP, the effect of capacity estimation errors, and the modifications proposed by XCP-b. § 4 elaborates on the problems of typical congestion control in multi-hop wireless networks. § 5 describes the simulation scenarios and presents the performance results. In § 6 we present the conclusions of the paper and discuss future work. In this paper we extend the validation of XCP-b to multihop wireless network scenarios. We evaluate the performance of XCP-b in simple chain-topologies, as well as in scenarios with mesh wireless access networks. The results show that XCP-b is viable in such scenarios, and that it outperforms regular TCP in terms of queuing delay, fairness and throughput stability. Such characteristics benefit primarily interactive and streaming applications which perform better with low latency and stable throughput. The results, however, show that throughput stability is much worser than in the case of single-hop wireless networks, mostly because in multi-hop 802.11 networks the hidden node problem may allow long bursts from single stations, and long idle periods for other stations. Another interesting aspect of XCP-b is that, unlike TCP, it maintains high efficiency when the capacity grows, making it suitable for next-generation 802.11n networks which claim to achieve speeds in the order of hundreds of Mbit/s.. We also study the sensitivity of the algorithm to the variation of its parameter values.ȃWe conclude that the values suggested in [3] achieve the best results, but some trade-offs exist. Namely the increase of the stabilizing queue length κ can increase total utilization at the expense of extra queuing delay. On the other hand, the value of τ can be lowered to reduce this queuing delay.. The experiments also show that the congestion windows produced by XCP-b are closer to the theoretical limit found in [5] , than the congestion window set by TCP Newreno. However, they are still above this limit as a result of queuing delay, which comes from the necessity of XCP-b buildup queues in the nodes. This increases competition for the medium, and produces sub-optimal spatial reuse.. We see as future work the incorporation of metrics, other than persistent queue variation into the feedback function of XCP-b. Such metrics could include the number of MAC collisions or MAC idle time, and they may remove the need to have queue build-up, removing the need to have the network in the saturation point. This approach may help to optimize spatial reuse in multi-hop wireless scenarios. In recent years, congestion control in wireless multi-hop networks has been the topic of several studies, including [9] , [8] , [5] , [13] , [6] , [14] . These studies focus on the interaction between TCP and the 802.11 MAC layer: the prob-lems of self-interference, capture, high latency, unfairness, and capacity limits. In [8] , it is noted that exists a value of the congestion window that maximizes throughput, by maximizing spatial reuse. Beyond this value, collisions and interference start to reduce the capacity. This value of the congestion window is generally much lower than the one at which TCP Newreno operates. [5] supports these results by showing that exists an upper-bound to the bandwidth delay product of a multi-hop wireless network. These studies provide a good source for comparison of the congestion window obtained by the congestion control algorithms, however, they only apply to the case of stand-alone multi-hop wireless networks.. In [17] the authors present an alternative XCP router feedback algorithm which can also be used in 802.11 networks. This algorithm differs from ours in the sense that it uses MAC level information, while we only rely on the queue dynamics.. In this paper we extend the validation and performance study of XCP-b to multi-hop wireless networks. We only give a brief explanation of the algorithm in this paper, for a more detailed description the reader should refer to [3] , [12] , and [7] .

O O O O O O O B I O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Approximate inference in Bayesian networks using binary probability trees The present paper introduces a new kind of representation for the potentials in a Bayesian network: binary probability trees. They enable the representation of context-specific independences in more detail than probability trees. This enhanced capability leads to more efficient inference algorithms for some types of Bayesian networks. This paper explains the procedure for building a binary probability tree from a given potential, which is similar to the one employed for building standard probability trees. It also offers a way of pruning a binary tree in order to reduce its size. This allows us to obtain exact or approximate results in inference depending on an input threshold. This paper also provides detailed algorithms for performing the basic operations on potentials (restriction, combination and marginalization) directly to binary trees. Finally, some experiments are described where binary trees are used with the variable elimination algorithm to compare the performance with that obtained for standard probability trees. Bayesian networks are graphical models that can be used to handle uncertainty in probabilistic expert systems. They provide an efficient representation of joint probability distributions. It is known that exact computation [1] of the posterior probabilities, given certain evidence, may become unfeasible for large networks. As a consequence, improved algorithms and methods are continuously proposed to enable exact inference on larger Bayesian networks. For example, in [2] it is presented an alternative method for improving the time required for accessing the values stored in potentials (and producing substantial savings in computation time when performing combination, marginalization or addition operations on them); the paper in [3] describes some improvements to message computation in Lazy propagation. Unfortunately, even with these improvements inference on complex Bayesian networks may be still unfeasible. This has led to the proposal of different approximate algorithms. These algorithms provide results in shorter time, albeit inexact. Some of the methods are based on Monte Carlo simulation, and others rely on deterministic procedures. Some of the deterministic methods use alternative representations for potentials, such as probability trees [4] [5] [6] . This representation offers the possibility to take advantage of context-specific independences. Probability trees can be pruned and converted into smaller trees when potentials are too large, thus facilitating approximate algorithms. In the present paper, we introduce a new kind of probability trees in which the internal nodes always have two children. They will be called binary probability trees. These trees allow the specification of fine-grained context-specific independences in more detail than standard trees, and should work better than standard probability trees for Bayesian networks containing variables with a large number of states.. The remainder of this paper is organized as follows: in Section 2 we describe the problem of probability propagation in Bayesian networks. Section 3 explains the use of probability trees to obtain a compact representation of the potentials and presents the related notation. In Section 4, we introduce binary probability trees and describe the procedure to build them from a potential, and how they can be approximated by pruning terminal trees; we also show the algorithms for direct application of the basic operations with potentials to binary probability trees. These algorithms are very similar to the algorithms for performing operations in mixed trees (trees with continuous and discrete variables) [7] . Section 5 provides details of the experimental work. Finally, Section 6 gives the conclusions and future work.

O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Fused, multi-spectral automatic target recognition with XCS We present new results from our most recent efforts in applying XCS to automatic target recognition (ATR). We place particular emphasis on ATR as a series of linked problems, which include pre-processing of multi-spectral data, detection of objects (in this case, vehicles) in that data, and identification (classification) of those objects. Multi-spectral data contains visual imagery, and additional imagery from several infrared spectral bands. The performance of XCS, with robust features, notably exceeds that of a template-based classifier on the pre-processed multi-spectral data for vehicle identification. This paper reviews our most recent efforts in using XCS in Automatic Target Recognition (ATR) 000. We are developing a vehicle identification scheme that includes registration, detection and classification modules, all of which are important components of a higher-level, end-to-end ATR scheme.

O O B I I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A class of Sturm-Liouville difference equations: (Non)oscillation constants and property BD Abstract-In this paper, we study oscillatory properties of the higher-order Sturm-Liouville difference equationIn particular, we derive sufficient conditions for (non)oscillation of (*) and compute explicitly its (non)oscillation constants.The obtained results are used to derive necessary and sufficient conditions for discreteness and boundedness below the spectrum of associated difference operators. We discuss also (non)oscillation of the Euler-type difference equation. In this paper, we investigate oscillatory behavior of the higher-order Strum-Liouville difference equation. (-l)"An (k@) Anyk) = q/cy/c+n,. where (II E R and kc") := I'(k + l)/I'(k -cx + 1) is the discrete factorial function, I'(t) := 0. DoSLP AND R. HILSCHER That is, equation (1) is viewed ss a perturbation of (2) . We establish also a nonoscillation criterion for a more general difference equation (-l) "A" (Q Any,9 = qk?/k+nr (3) where rk > 0. As an application, we determine the (non)oscillation of the Euler-type difference equation (-l) "An (,(-I A%) = &Yk+n, (4 i.e., (I) with qk = X/k (2n-a), X E R, which is the limiting case in our criteria. The results are also used to derive necessary and sufficient conditions for discreteness and boundedness below (property BD) th e spectrum of associated singular difference operators B(Y)k+n = (-1) k n (01) A" (rk A"Yk). in the weighted Hilbert space e, = {y = {yk}~=r, C" k-(a)yz < co}, where km(") = l/k(a), (Non)oscillation and BD criteria given in this work may be regarded as the discrete analogue of criteria for the corresponding Sturm-Liouville differential equation. (-1)" (t" y'"') (n) = q(t)y and for the singular differential operator P(y) E (-lyta (r(t)y("y").. Although we extend and complete criteria for difference equations, the method gives directly a way to new results also for differential equations. A general background of the spectral theory of linear operators in a Hilbert space can be found in the monograph [l] , and the specialization of this theory to differential and difference operators is given in . Concerning the basic facts of oscillation and spectral theory of the secondorder Sturm-Liouville difference operators, we refer to [5] . The results of that paper have been extended by several authors to higher-order difference operators in [6-lo] . In these papers, it was shown that oscillation and spectral properties of difference operators are similar to those of their differential counterparts, even if the investigation of difference operators requires frequently more effort to overcome technical difficulties (mostly caused by the discrepancies between differential and difference calculus), which do not occur in the continuous case.. Becall that oscillation and spectral properties of differential operators K(y) z $$ (r(t)y'^')("). (with a particular attention devoted to the case w(t) = tsa', cr E JR) were investigated in [8, . In these papers, (non)oscillation criteria for the equation (-1)" (r(t)P)(") = w(t)y are derived, and associated conditions for discreteness and boundedness below the spectrum of K are given in terms of the relationship between the functions r, w. Our results concerning equation (1) also follow this line. We show that if the sequence qk in (1) is not eventually too positive (it is eventually sufficiently positive) with respect to the sequence k("), then (1) is nonoscillatory (oscillatory). More precisely, our investigation is closely related to the socalled conditionally oscillatory equations and their oscillation constants; for details see Section 5.. Note also that (non)oscillation criteria for differential equation (6) given in the above-mentioned papers are in many cases based on the comparison of this equation with the Euler-type differential equation (-l)n(tay(n))(n) = (X/tzn--a )y, whose oscillation behavior depends on the value of the constant X. However, this approach does not extend directly to difference equation (4), since the discrete Euler equation is not in a self-adjoint form; see [23, Chapter 31. For this reason, we needed to use a new method, based on the factorization of certain difference operators, and we are able to determine (non)oscillation of the selfadjoint Euler-type difference equation (4) for some intervals of X. This method yields also new results when applied to differential equations (6); see Section 6.. The paper is organized as follows. In the next section, we collect auxiliary material. We mention the relationship between higher-order Sturm-Liouville difference equations and linear Hamiltonian difference systems. We recall the connection between oscillation and spectral properties of difference operators as well. We also give basic facts concerning linear difference operators, in particular, their factorization and adjoint operators. In Section 3, we present nonoscillation criteria for equation (3) (Theorem 1) and for equation (1) (Theorems 2-5). These criteria are based on a self-adjoint factorization of the one-term difference operator An(kca) Anyk), which is explicitly computed in Lemmas 3-5. Oscillation criteria for equation (1) are derived in Section 4 (Theorems 6-8). Both types of criteria involve computing an explicit form of the (non)oscillation constants. Section 5 contains applications to Euler-type difference equation (4) (Theorems 9 and 10) and to BD criteria for the spectrum of the difference operator a (Theorems 11-16). Throughout the paper, we formulate several conjectures deserving further research. Section 6 is devoted to remarks concerning the results of this paper and their possible extensions, and in Section 7 we collect technical results needed in our computations.

O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Bias and scalability in evolutionary development The introduction of a genotype-phenotype map modelled on biological development can potentially improve the scalability of evolutionary algorithms. Previous work by Gordon and Bentley demonstrated that such a model can be used to evolve patterns that map to useful but small phenotypes. This paper uses the same model to generate much larger patterns covering arrays of up to 64x64 cells. The results show that the model's performance is generally comparable to similar development-based systems [12, 14] , and with some measures outperforms them. Additionally the inherent biases of the model are explored, such as the need to use symmetry-breaking initial conditions which some other models do not require. This exploration yields a set of guidelines that suggest what kinds of problem the model is suited to exploring. It is widely recognized that the inability of evolutionary algorithms to scale to large and complex problems is a major impediment to their application in the real world [8, 9, 17] .. In the past few years a number of researchers have begun to explore how modelling the genotype-phenotype map on biological development might improve scalability [8, 12, 14] . In [6] Gordon and Bentley introduced such a model and in [5] it was demonstrated that it could be used to evolve patterns that mapped to useful phenotypes, in this case circuit designs. However the patterns and consequently the circuits they mapped to were small. This paper shows that the same model can be used to evolve high fitness solutions for much larger pattern generation problems, which is important for the generation of large phenotypes.. Recently other researchers have presented evolutionary systems that incorporate various models of development and have applied them to similar pattern generation problems. This paper compares Gordon and Bentley's system with these other systems. The results show that the model used by Gordon and Bentley provides comparable performance to the best of these.. Although genotype-phenotype mappings based on development have the potential to benefit scalability they apply strong biases that affect the nature of the evolved phenotypes. This paper presents an analysis of the biases inherent to Gordon and Bentley's model and derives a set of guidelines that can be used to determine what kinds of problem the model is suited to.. The rest of this paper is structured as follows: section 2 presents a summary of Gordon and Bentley's developmental model. Section 3 explores the generation of large patterns and derives guidelines for determining if this model might be suited to a particular problem. Section 4 presents the evolution of more complex problems, and compares the performance of Gordon and Bentley's model to others found in the literature, and provides possible explanations for the observed greater performance of the Gordon and Bentley model. Conclusions are made in Section 5. The problem of scalability is one of great importance if evolutionary techniques are to be routinely applied to large realworld problems, such as digital circuit design. One possible solution is to use a developmental genotype-phenotype mapping.. However research into such systems is at an early stage: they are poorly understood and scalability has not been widely demonstrated through experiment. This paper has shown through a number of experiments that the developmental model used in [5] is capable of generating large phenotypes. In future work it is intended to apply the model to the evolution of large digital circuits. Through these experiments a number of rules have been derived that suggest what kinds of problem are more suited to such an approach. The developmental model used in this paper has also been compared to similar models in the literature.. Although it provides more domain knowledge than some of these other models, it appears to provide at least comparable, and in many cases better performance for the problems presented here while using fewer evaluations.

O O B O O B O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A global wiring paradigm for deep submicron design Abstract-Global interconnect is commonly regarded as a key potential bottleneck to the advancing performance of high-speed integrated circuits. Our previous work has suggested that local interconnect effects can be managed through a deep submicron design hierarchy that uses 50 000 to 100 000 gate modules as primitive building blocks. The primary goal of this paper is to examine global interconnect effects, within such a design hierarchy, to determine if there are any significant roadblocks which will prevent National Technology Roadmap for Semiconductors (NTRS) performance expectations from being met. Specifically, the issues of global resistance-capacitance delay, signal time-of-flight, inductance, clock and power distribution, and noise are studied. Results indicate that, while global clock frequencies will necessarily be lower than local clock speeds, NTRS expectations should be attainable to the 50-nm technology generation. Achieving these high clock speeds (10-GHz local clock) will be aided by the use of a newly proposed routing hierarchy which limits interconnect effects at each level of a design (local, isochronous, and global).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O


Direct trimming of NURBS surfaces on the GPU This paper presents a highly efficient direct trimming technique for NURBS surfaces, which is applicable to tessellation-based rendering as well as ray tracing systems. The central idea is to split the trim curves into monotonic segments with respect to the two parameter dimensions of the surface patches. We use an optimized bisection method to classify a point with respect to each monotonic trim curve segment without performing an actual intersection test. Our hierarchical acceleration structure allows the use of a large number of such curve segments and performs the bisection method only for points contained in the bounding boxes of the curve segments.We have integrated our novel point classification scheme into a GPU-based NURBS ray casting system and implemented the entire trimmed NURBS rendering algorithm in a single OpenGL GLSL shader. The shader can handle surfaces and trim curves of arbitrary degrees, which allows the use of original CAD data without incorporating any approximations. Performance data confirms that our trimming approach can deal with hundreds of thousands of trim curves at interactive rates. Our point classification scheme can be applied to other application domains dealing with complex curved regions including flood fills, font rendering and vector graphics mapped on arbitrary surfaces. Computer Aided Design (CAD) modeling tools are widely used for the industrial design of models for prototyping and production. The standard surface representation within these systems are trimmed non-uniform rational B-Spline (NURBS) surfaces. Trimmed NURBS surfaces are capable of compactly describing almost any shape and additionally provide local and intuitive control. These patches are commonly visualized by rendering a trianglebased approximation (tessellation) on graphic processing units (GPU), since there is no specialized NURBS rendering hardware readily available. The process of generating a high-quality tessellation is time consuming, especially for trimmed surfaces requiring fine sampling of the trim boundaries. Recent on-the-fly tessellation techniques propose trimming approaches for the CPU [Balázs et al. 2004] as well as for the GPU [Guthe et al. 2005] . However, since they either use mesh-based or texture-based techniques, updates of these representations are required to maintain pixel-accuracy during interactive viewpoint manipulations.. In this paper we present a highly efficient direct trimming technique for NURBS surfaces, which is applicable to tessellation-based rendering as well as ray tracing systems. Trimming is accomplished by employing the ray-based point-in-curve test with respect to the NURBS trim curves. Our central idea is to split the trim curves into monotonic segments such that there is at most a single intersection with a horizontal ray. The split points correspond to the extrema of the trim curves and can easily be found. Within each single-intersection curve segment we use an optimized bisection method to classify a point as being inside or outside of the trim region without performing an actual ray-curve intersection. The set of monotonic trim curve segments for each patch suggest a two-level hierarchical acceleration structure, which organizes the entire trim region into cells containing, in most cases, one trim curve segment or none at all. During NURBS rendering the acceleration structure is searched for the cell containing the point to be classified and only the contained trim curve segments are further considered with our optimized bisection method.. Common trimming implementations use Bézier Clipping as a robust technique for finding all the intersections of a ray and a trim curve. However, Bézier Clipping is originally a recursive technique, which is difficult to efficiently implement on a GPU. An iterative GPU implementation of Bézier Clipping ( [Pabst et al. 2006] ) was limited to a small number of curves due to GPU programming constraints and the lack of an acceleration structure. Furthermore, Bézier Clipping requires quite a number of computationally nontrivial steps such as convex hull computations and Bézier polygon subdivisions. A robust implementation also needs to consider numerous special cases ( [Wang et al. 2000; Efremov et al. 2005] ).. The main contribution of this paper is a new point classification scheme for curved regions with holes. We apply this scheme for efficiently handling a large number of trim curves in real-time rendering systems. Our approach does not use any approximations of the trim curves, nor is the degree of the curves limited. Due to our trimcurve preprocessing we are able to employ an optimized bisection method for classifying a point with respect to a bi-monotonic curve segment. Problematic ray-curve intersection cases such as multiple zeros, minima and maxima are resolved during a preprocessing step, all of which contribute to the robustness of our method. A complexity estimation conveys that our classification of a point contained in the bounding box of a trim curve segment requires on average less than two evaluations of the curve. This estimation was also empirically confirmed. A comparison reveals that the commonly employed Bézier Clipping technique is about five times slower than our technique for the investigated real world examples. However, due to the precision of our approach, inaccuracies of the actual trim curve representations may be revealed, which results in pin holes or cracks along the trim boundary of two adjacent surface patches. For evaluation purposes, we integrated our trimming method into a GPU-based ray casting system. The entire rendering algorithm for trimmed NURBS surfaces is implemented as a single fragment program which can handle surfaces and trimming curves of arbitrary degree. Due to our two-level acceleration structure we achieve interactive frame rates for models containing more than a hundred thousand trim curves. The efficiency of our approach makes it applicable to a variety of other domains requiring point classification with respect to complex curved regions including font rendering, flood filling, and vector graphics mapped on arbitrary surfaces. In this paper we presented a novel point classification method for curved regions with holes and applied this technique to direct trimming of NURBS surfaces. We demonstrated how to split the original trimming curves into segments such that the commonly used computationally expensive Bézier Clipping method can be replaced by an adapted bisectioning algorithm. Furthermore, we make use of the fact that the intersection point for the ray-based point-in-closedcurve test itself is not necessary. Our optimized bisectioning algorithm requires on average less than two trimming curve evaluations to classify a point contained in the curve's bounding box. A highly adapted acceleration data structure effectively limits the intersection candidates such that we can deal with complex trimming regions. Our point classification implementation supports trimming curves of arbitrary degree. It is integrated into a GPU-based single pass NURBS ray casting system, which is realized as a single OpenGL GLSL program. Our experience with the system shows that complex real-world models can be rendered at interactive frame rates.. Nevertheless, there is still room for improving the trimming approach. In particular, the combination with a coarse (u, v)-grid could avoid unnecessary binary searches in our hierarchical acceleration structure. If the lookup into the grid indicates the cell to be completely inside or outside the trimming domain no further computations have to be done. Otherwise the regular algorithm has to be used. Furthermore, we could further split the trimming curves to reach a statistically optimal computation time for the point-inclose curve test under the assumption of an even distribution of the point tests over the (u, v)-domain. The cost for a trimming curve evaluation is dominated by the number of texture fetches of the coefficients and thus known during the preprocessing phase. Since we know that we need on average about two curve segment evaluations for points in the bounding box of a curve segment, the overall influence on the number of texture fetches by introducing a split can be computed. The binary searches also need to be considered in this cost function. Their cost is dominated by the lookup of the v-and u-interval boundaries. Thus an optimization of the overall cost will create an optimal split of the trim curve segments under the given assumptions.. As pointed out by [Song et al. 2004] , it is a hopeless task to exactly represent the intersection of two NURBS patches. This exactness is not essential for most applications, provided that trimmed surfaces agree exactly along approximate intersection curves without gaps or overlaps between them. They suggest control point perturbation schemes to enforce watertight patch intersections. However, this suggestion has not yet found its way into classical CAD systems as we found out: with large magnifications of adjacent trimmed patches, we sometimes find small pinholes along the trim curves or with even larger magnifications, gaps occur. While this problem should be resolved within the CAD systems, it could be also handled by the rendering system. One could even allow the user to switch between the two visualizations.. The most important next step is the integration of our approach into a CAD system. The small acceleration data structure can be generated and downloaded into the graphics card on the fly while trim curves are manipulated. Thus, the system can always deal with a pixel-precise representation of the trimming curves. For a large number of NURBS patches the performance of our ray-castingbased surface rendering algorithm is still limited on current GPUs, but the addition of occlusion culling techniques and the use of an early-ray termination test will significantly improve performance for scenes with medium to high depth complexity. We are convinced that direct GPU-based rendering of trimmed NURBS surfaces is a worthwhile alternative to the current tessellation-based rendering pipeline. Trimming is just one, albeit very useful, application of our point classification technique. It could also be used for other applications where point classification with respect to curved regions with holes is required, including collision detection, stenciling, flood fills and vector graphics mapped onto arbitrary surfaces. We provide a short overview of existing rendering techniques for trimmed NURBS surfaces with a focus on the applied trimming method. The surface rendering techniques can be classified as methods based on tessellation or on ray tracing.. Tessellation-based approaches convert the NURBS surface representation into a set of triangles. [Rockwood et al. 1989] and [Kumar and Manocha 1995] use uniform subdivision methods to generate piece-wise linear approximations of the trimming curves, and use them as polygonal boundaries for the surface tessellation. [Balázs et al. 2004] propose an elevation of the trimming curves into Euclidian space to constrain the error of a piece-wise linear approximation of the trim curves.. In contrast to these preprocessing-based meshing methods, [Guthe et al. 2005] adaptively triangulate the NURBS surface on the GPU and perform the actual trimming task on a per-pixel basis using a lookup-texture. They generate a texture representation of the trimming area by rendering a polygonal representation of each trim loop. The resolution of the trim texture representations is adapted to the resolution of the trimmed surface on the screen and updated on the fly if necessary. All of these approaches use a polygonal approximation of the trim curves and a tessellation of the NURBS surfaces. Furthermore, for high screen resolutions these techniques need to create an enormous number of triangles and a large number of high resolution trim textures, both of which may need to be continuously updated during viewpoint navigation. While these techniques may produce similar quality images as our approach, we require significantly less memory on the GPU since only the control points of the trimming curves and a small acceleration structure need to be stored and potentially updated during surface editing.. Pioneering work in the field of parametric surface ray tracing [Kajiya 1982; Toth 1985; Joy and Bhetanabhotla 1986] focuses on different ray-surface intersection methods, but does not provide a solution for the trimming problem. [Farouki 1987 ] was the first to introduce a set of generalized algorithms for processing trimmed surfaces. The seminal paper on "Ray Tracing Trimmed Rational Surface Patches" by [Nishita et al. 1990 ] presented the Bézier Clipping method as a robust numerical root-finding algorithm, which can be used for ray-surface intersections as well as for ray-curve intersections. It exploits the convex hull property of Bézier representations and recursively subdivides curve segments containing potential roots. [Wang et al. 2000] and [Efremov et al. 2005] proposed several enhancements to further improve the stability and applicability of the Bézier Clipping algorithm.. Several NURBS ray tracing systems, including [Geimer and Abert 2005; Benthin et al. 2004; Efremov et al. 2005] , make use of Bézier Clipping for trimming. [Efremov et al. 2005 ] utilizes an adapted version of the ray-casting-based point-in-polygon test for trim loops. They apply Bézier Clipping for intersecting a ray with the trim curves. Additionally, a trim loop hierarchy accelerates the trimming test by avoiding unnecessary intersections. [Benthin et al. 2004; Geimer and Abert 2005] also use the point-in-polygon test and Bézier Clipping. They almost reach interactive frame rates due to the clever use of SIMD extensions and Kd-tree-based acceleration structures. However, they only deal with cubic trimming curves.. [ Pabst et al. 2006 ] presented a GPU-based NURBS ray casting system. They provided an iterative formulation of the originally recursive Bézier Clipping algorithm to directly perform the trimming task on the GPU during fragment processing. However, the system was limited to a small number of trimming curves of low degree due to several constraints of current graphics hardware and the lack of an acceleration structure.. Splitting curves into monotonic segments is used in many works. The idea of preprocessing curves into bi-monotonic segments was introduced by [Mudur and Koparkar 1984] for use in an algorithm for intersecting two planar parametric curves. [Rockwood et al. 1989 ] split trim curves into monotonic segments to robustly generate the tessellation of trimmed NURBS patches. For this process, they also developed their own root finder, which has some similarities to Bézier Clipping. [Keyser et al. 1999 ] base their MAPC library for manipulating algebraic curves (e. g. intersecting two curves) on monotonic segments. They call the process of splitting a curve into monotonic segments "resolving the curve topology". [Qin et al. 2008 ] split curves into monotonic segments to simplify distance computations.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters We present the Mathematica package HypExp which allows to expand hypergeometric functions J FJ−1 around integer parameters to arbitrary order. At this, we apply two methods, the first one being based on an integral representation, the second one on the nested sums approach. The expansion works for both symbolic argument z and unit argument. We also implemented new classes of integrals that appear in the first method and that are, in part, yet unknown to Mathematica. As solutions of a large class of differential equations, hypergeometric functions P F Q appear in many branches of science. They appear, in particular, in particle physics during the calculation of radiative corrections to scattering cross sections in loop [1] [2] [3] [4] [5] [6] or phase space [7] [8] [9] integrals. In the context of dimensional regularisation, the arguments of the hypergeometric functions have to be expanded in a small parameter around integer or rational arguments.. Until recently, there was no systematic approach to the expansion of hypergeometric functions. The required expansions have been produced with a case-by-case approach. Recently a general algorithm has been developed [10] for expanding hypergeometric functions and other transcendental functions systematically around their parameters. This algorithm was implemented [11] in the framework of GiNaC [12] . Related work was also presented in Refs. [13, 14] .. Until now, an implementation of the expansion of hypergeometric functions around their parameters was missing in the widely used computer algebra systems Mathematica [15] and Maple [16], allowing to use hypergeometric functions in connection with the multi-purpose features of these programs.. The aim of this work is to provide an implementation of these expansions for Mathematica. This implementation is prepared in the form of a Mathematica package that aims to be tunable and user-friendly. This paper is structured as follows. In the next two sections, we present the two approaches we used in the package. Section 4 is devoted to hypergeometric functions of unit argument. In section 5 we describe the implementation of our methods in the package HypExp and provide examples of its usage. Finally, we conclude with a summary. In this paper we presented the Mathematica package HypExp for expanding arbitrary hypergeometric functions to arbitrary order in a small quantity around integer-valued parameters. These expansions are required for example in the computation of multi-loop or multi-particle phase space integrals in dimensionally regularized quantum field theory.. A first application was presented in ref. [6] , namely the ǫ-expansion of the two-loop quark and gluon form factors whose exact analytic expressions contain hypergeometric functions in addition to Γ-functions.

O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O


The minimum-rank gram matrix completion via modified fixed point continuation method The problem of computing a representation for a real polynomial as a sum of minimum number of squares of polynomials can be casted as finding a symmetric positive semidefinite real matrix of minimum rank subject to linear equality constraints. In this paper, we propose algorithms for solving the minimum-rank Gram matrix completion problem, and show the convergence of these algorithms. Our methods are based on the fixed point continuation method. We also use the Barzilai-Borwein technique and a specific linear combination of two previous iterates to accelerate the convergence of modified fixed point continuation algorithms. We demonstrate the effectiveness of our algorithms for computing approximate and exact rational sum of squares decompositions of polynomials with rational coefficients. This section provides implementation details of the modified FPC algorithm for solving the minimum-rank Gram matrix completion problem.

O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O B I I O