Abstract: The multiple-sets split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator's range. It generalizes the convex feasibility problem as well as the two-sets split feasibility problem. We propose a projection algorithm that minimizes a proximity function that measures the distance of a point from all sets. The formulation, as well as the algorithm, generalize earlier work on the split feasibility problem. We offer also a generalization to proximity functions with Bregman distances. Application of the method to the inverse problem of intensity-modulated radiation therapy treatment planning is studied in a separate companion paper and is here only described briefly.

Abstract: We present ONETEP (order-N electronic total energy package), a density functional program for parallel computers whose computational cost scales linearly with the number of atoms and the number of processors. ONETEP is based on our reformulation of the plane wave pseudopotential method which exploits the electronic localization that is inherent in systems with a nonvanishing band gap. We summarize the theoretical developments that enable the direct optimization of strictly localized quantities expressed in terms of a delocalized plane wave basis. These same localized quantities lead us to a physical way of dividing the computational effort among many processors to allow calculations to be performed efficiently on parallel supercomputers. We show with examples that ONETEP achieves excellent speedups with increasing numbers of processors and confirm that the time taken by ONETEP as a function of increasing number of atoms for a given number of processors is indeed linear. What distinguishes our approach is that the localization is achieved in a controlled and mathematically consistent manner so that ONETEP obtains the same accuracy as conventional cubic-scaling plane wave approaches and offers fast and stable convergence. We expect that calculations with ONETEP have the potential to provide quantitative theoretical predictions for problems involving thousands of atoms such as those often encountered in nanoscience and biophysics.

Abstract: We extend the rotation measure work of Burn (1966) to the cases of limited sampling of lambda squared space and non-constant emission spectra. We introduce the rotation measure transfer function (RMTF), which is an excellent predictor of n-pi ambiguity problems with the lambda squared coverage. Rotation measure synthesis can be implemented very efficiently on modern computers. Because the analysis is easily applied to wide fields, one can conduct very fast RM surveys of weak spatially extended sources. Difficult situations, for example multiple sources along the line of sight, are easily detected and transparently handled. Under certain conditions, it is even possible to recover the emission as a function of Faraday depth within a single cloud of ionized gas. Rotation measure synthesis has already been successful in discovering widespread, weak, polarized emission associated with the Perseus cluster (De Bruyn and Brentjens, 2005). In simple, high signal to noise situations it is as good as traditional linear fits to polarization angle versus lambda squared plots. However, when the situation is more complex or very weak polarized emission at high rotation measures is expected, it is the only viable option.

Abstract: This pai>er views the standard production function in macroeconomics as a reduced form and derives its properties from microfoundations. The shape of this production function is governed by the distribution of ideas. If that distribution is Pareto, then two results obtain: the global production function is Cobb-Douglas, and technical change in the long run is labor-augmenting. Kortum showed that Pareto distributions are necessary if search-based idea models are to exhibit steady-state growth. Here we show that this same assumption delivers the additional results about the shape of the production function and the direction of technical change.

Abstract: An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.

Abstract: This paper describes ZEUS-MP, a multi-physics, massively parallel, message- passing implementation of the ZEUS code. ZEUS-MP differs significantly from the ZEUS-2D code, the ZEUS-3D code, and an early "version 1" of ZEUS-MP distributed publicly in 1999. ZEUS-MP offers an MHD algorithm better suited for multidimensional flows than the ZEUS-2D module by virtue of modifications to the Method of Characteristics scheme first suggested by Hawley and Stone (1995), and is shown to compare quite favorably to the TVD scheme described by Ryu et. al (1998). ZEUS-MP is the first publicly-available ZEUS code to allow the advection of multiple chemical (or nuclear) species. Radiation hydrodynamic simulations are enabled via an implicit flux-limited radiation diffusion (FLD) module. The hydrodynamic, MHD, and FLD modules may be used in one, two, or three space dimensions. Self gravity may be included either through the assumption of a GM/r potential or a solution of Poisson's equation using one of three linear solver packages (conjugate-gradient, multigrid, and FFT) provided for that purpose. Point-mass potentials are also supported. Because ZEUS-MP is designed for simulations on parallel computing platforms, considerable attention is paid to the parallel performance characteristics of each module. Strong-scaling tests involving pure hydrodynamics (with and without self-gravity), MHD, and RHD are performed in which large problems (256^3 zones) are distributed among as many as 1024 processors of an IBM SP3. Parallel efficiency is a strong function of the amount of communication required between processors in a given algorithm, but all modules are shown to scale well on up to 1024 processors for the chosen fixed problem size.

Abstract: The classical Duhem model provides a finite-dimensional differential model of hysteresis. In this paper, we consider rate-independent and rate-dependent semilinear Duhem models with provable properties. The vector field is given by the product of a function of the input rate and linear dynamics. If the input rate function is positively homogeneous, then the resulting input-output map of the model is rate independent, yielding persistent nontrivial input-output closed curve (that is, hysteresis) at arbitrarily low input frequency. If the input rate function is not positively homogeneous, the input-output map is rate dependent and can be approximated by a rate-independent model for low frequency inputs. Sufficient conditions for convergence to a limiting input-output map are developed for rate-independent and rate-dependent models. Finally, the reversal behavior and orientation of the rate-independent model are discussed.

Abstract: A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q-proper-hypergeometric function, and thus it is q-holonomic. We demonstrate our results by computer calculations.

Abstract: The generalized knapsack function is defined as f a (x)= Σ i a i x i , where a = (a 1 ,...,a m ) consists of m elements from some ring R, and x = (x i ,...,x m ) consists of m coefficients from a specified subset S C R. Micciancio (FOCS 2002) proposed a specific choice of the ring R and subset S for which inverting this function (for random a, x) is at least as hard as solving certain worst-case problems on cyclic lattices. We show that for a different choice of S ⊂ R, the generalized knapsack function is in fact collision-resistant, assuming it is infeasible to approximate the shortest vector in n-dimensional cyclic lattices up to factors 0(n). For slightly larger factors, we even get collision-resistance for any m > 2. This yields very efficient collision-resistant hash functions having key size and time complexity almost linear in the security parameter n. We also show that altering S is necessary, in the sense that Micciancio's original function is not collision-resistant (nor even universal one-way). Our results exploit an intimate connection between the linear algebra of n-dimensional cyclic lattices and the ring Z[α]/(α n -1), and crucially depend on the factorization of a n - 1 into irreducible cyclotomic polynomials. We also establish a new bound on the discrete Gaussian distribution over general lattices, employing techniques introduced by Micciancio and Regev (FOCS 2004) and also used by Micciancio in his study of compact knapsacks.

Abstract: The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h m ,L) of the maximal height h m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h m ,L)=L−1/2f(h m L−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(h m ,L)=L−1/2F(h m L−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].

Abstract: A user is able to access additional functions not represented in a current image displayed by a graphical user interface. At least one function not presented on the current image is represented by a symbol on an underlying image that is at least partially covered by the current image. When the user performs a predetermined user input (e.g., selecting a corner of the current image), the underlying image and the at least one function represented thereby become accessible. When the user input is performed, a visual effect depicts the current image being at least partially removed from over the underlying image, thereby revealing and permitting access to the at least one additional function. The user input is made by the user performing an action with the user's hand or another object adjacent to a responsive display, or by using a pointing device to manipulate a displayed image.

Abstract: We investigate the problem of model selection for learning algorithms depending on a continuous parameter. We propose a model selection procedure based on a worst-case analysis and on a data-independent choice of the parameter. For the regularized least-squares algorithm we bound the generalization error of the solution by a quantity depending on a few known constants and we show that the corresponding model selection procedure reduces to solving a bias-variance problem. Under suitable smoothness conditions on the regression function, we estimate the optimal parameter as a function of the number of data and we prove that this choice ensures consistency of the algorithm.

Abstract: Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective.

Abstract: The main purpose of this paper is to present a systemic study of some families of multiple $q$-Euler numbers and polynomials. In particular, by using the $q$-Volkenborn integration on $\Bbb Z_p$, we construct $p$-adic $q$-Euler numbers and polynomials of higher order. We also define new generating functions of multiple $q$-Euler numbers and polynomials. Furthermore, we construct Euler $q$-Zeta function. This paper is the version of the preprint of my paper to publish in the special issues as the preprint part of this proceedings.

Abstract: We report a new algorithm for constructing pathways between local minima that involve a large number of intervening transition states on the potential energy surface. A significant improvement in efficiency has been achieved by changing the strategy for choosing successive pairs of local minima that serve as endpoints for the next search. We employ Dijkstra's algorithm to identify the `shortest' path corresponding to missing connections within an evolving database of local minima and the transition states that connect them. The metric employed to determine the shortest missing connection is a function of the minimised Euclidean distance. We present applications to the formation of buckminsterfullerene and to the folding of the B1 domain of protein G, tryptophan zippers, and the villin headpiece subdomain. The corresponding pathways contain up to 163 transition states, and will be used in future discrete path sampling calculations.

Abstract: New general purpose ranking functions are discovered using genetic programming. The TREC WSJ collection was chosen as a training set. A baseline comparison function was chosen as the best of inner product, probability, cosine, and Okapi BM25. An elitist genetic algorithm with a population size 100 was run 13 times for 100 generations and the best performing algorithms chosen from these. The best learned functions, when evaluated against the best baseline function (BM25), demonstrate some significant performance differences, with improvements in mean average precision as high as 32% observed on one TREC collection not used in training. In no test is BM25 shown to significantly outperform the best learned function.

Abstract: In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain f~ c R n, n >/1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of Au + / 3 ( u ) = f with/3 a nondecreasing function R ~ R , /3(0)=0, and f > ~ 0 a.e. in ~ if and only if the integral f ( / 3 ( s ) s ) l / 2 d s diverges at = 0 + . We extend the result to general S more equations, in particular to Ap u +/3 (u) = f where Ap (u) = div( I Du ]P 2Du), 1 < p < ~ . Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

Abstract: In this note, we present a model predictive control (MPC) algorithm for uncertain time-varying systems with input constraints and state-delay. Uncertainty is assumed to be polytopic, and delay is assumed to be unknown but with a known upper bound. For a memoryless state-feedback MPC law, we define an optimization problem that minimizes a cost function and relaxes it to two other optimization problems by finding an upper bound of the cost function. One is solvable and the other is not. We prove equivalence and feasibilities of the two optimization problems under a certain assumption on the weighting matrix. Based on these properties and optimality, we show that feasible MPC from the optimization problems stabilizes the closed-loop system. Then, we present an improved MPC algorithm that includes relaxation procedures of the assumption on the weighting matrix and stabilizes the closed-loop system. Finally, a numerical example illustrates the performance of the proposed algorithm.

Abstract: We describe an algorithm for the Feedback Vertex Set problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(ckn3) where c=10.567 and n is the number of vertices in the graph. The best previous algorithms were based on the method of bounded search trees, branching on short cycles. The best previous running time of an FPT algorithm for this problem, due to Raman, Saurabh and Subramanian, has a parameter function of the form 2O( klogk / loglogk). Whether an exponentially linear in k FPT algorithm for this problem is possible has been previously noted as a significant challenge. Our algorithm is based on the new FPT technique of iterative compression. Our result holds for a more general “annotated” form of the problem, where a subset of the vertices may be marked as not to belong to the feedback set. We also establish “exponential optimality” for our algorithm by proving that no FPT algorithm with a parameter function of the form O(2o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, namely FPT =M[1].

Abstract: In this paper we consider sampling based fitted value iteration for discounted, large (possibly infinite) state space, finite action Markovian Decision Problems where only a generative model of the transition probabilities and rewards is available. At each step the image of the current estimate of the optimal value function under a Monte-Carlo approximation to the Bellman-operator is projected onto some function space. PAC-style bounds on the weighted Lp-norm approximation error are obtained as a function of the covering number and the approximation power of the function space, the iteration number and the sample size.

Abstract: We are concerned with the semilinear differential equation in a Banach space X, x'(t)=Ax(t)+F(t,x(t)), t∈R, where A generates an exponentially stable C 0 -semigroup and F(t,x): R x X → X is a function of the form F(t,x) = P(t)Q(x). Under appropriate conditions on P and Q, and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.

Abstract: We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus The critical values of this function are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets Our algorithm is based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th degree univariate polynomial whose real roots give us one component of the critical points We discuss also some degenerate cases and show several examples, involving the orbits of the known asteroids and comets

Abstract: We propose an extension of map algebra to three dimensions for spatio-temporal data handling. This approach yields a new class of map algebra functions that we call "cube functions." Whereas conventional map algebra functions operate on data layers representing two-dimensional space, cube functions operate on data cubes representing two-dimensional space over a third-dimen- sional period of time. We describe the prototype implementation of a spatio-temporal data structure and selected cube function versions of conventional local, focal, and zonal map algebra functions. The utility of cube functions is demonstrated through a case study analyzing the spatio-temporal variability of remotely sensed, southeastern U.S. vegetation character over various land covers and during different El Nino/Southern Oscillation (ENSO) phases. Like conventional map algebra, the application of cube functions may demand significant data preprocessing when integrating diverse data sets, and are subject to limitations related to data storage and algorithm performance. Solutions to these issues include extending data compression and computing strategies for calculations on very large data volumes to spatio-temporal data handling.

Abstract: We discuss the stability of the Parareal algorithm for an autonomous set of differential equations. The stability function for the algorithm is derived, and stability conditions for the case of real eigenvalues are given. The general case of complex eigenvalues has been investigated by computing the stability regions numerically.