Intermediate Sums Research Articles

Parameterized Splitting of Summed Volume Tables

AbstractSummed Volume Tables (SVTs) allow one to compute integrals over the data values in any cubical area of a three‐dimensional orthogonal grid in constant time, and they are especially interesting for building spatial search structures for sparse volumes. However, SVTs become extremely memory consuming due to the large values they need to store; for a dataset of n values an SVT requires 𝒪(n log n) bits. The 3D Fenwick tree allows recovering the integral values in 𝒪(log3 n) time, at a memory consumption of 𝒪(n) bits. We propose an algorithm that generates SVT representations that can flexibly trade speed for memory: From similar characteristics as SVTs, over equal memory consumption as 3D Fenwick trees at significantly lower computational complexity, to even further reduced memory consumption at the cost of raising computational complexity. For a 641 × 9601 × 9601 binary dataset, the algorithm can generate an SVT representation that requires 27.0GB and 46 · 8 data fetch operations to retrieve an integral value, compared to 27.5GB and 1521·8 fetches by 3D Fenwick trees, a decrease in fetches of 97%. A full SVT requires 247.6GB and 8 fetches per integral value. We present a novel hierarchical approach to compute and store intermediate prefix sums of SVTs, so that any prescribed memory consumption between 𝒪(n) bits and 𝒪(n log n) bits is achieved. We evaluate the performance of the proposed algorithm in a number of examples considering large volume data, and we perform comparisons to existing alternatives.

Effects of Working Memory, Strategy Use, and Single-Step Mental Addition on Multi-Step Mental Addition in Chinese Elementary Students.

The aim of this paper was to examine the roles of working memory, single-step mental addition skills, and strategy use in multi-step mental addition in two independent samples of Chinese elementary students through different approaches to manipulate two dimensions of task characteristics (the primary task). In Study 1, we manipulated strategy types through the dimension of schema automaticity (whether intermediate sums were 10s) and the dimension of working memory load (WML, two steps versus four steps). A hierarchical linear model (HLM) analysis was conducted at case level, strategy level, and individual level. In Study 2, we manipulated task characteristics through schema automaticity (one-time versus two-time regrouping) and the WML (partial versus complete decomposition). A three-level HLM analysis was applied. The general findings of Study 1 and Study 2 suggested that shorter response time on single-step mental addition corresponded to shorter response time on multi-step mental addition. The use of strategies (from easier to more difficult strategies) negatively predicted response time on multi-step mental addition. Easier strategy was associated with shorter response time on multi-step mental addition. Better phonological loop was associated with shorter response time on multi-step mental addition. The findings in both studies highlighted the important role of phonological loop in mental addition in Chinese children, suggesting that the involvement of a specific subcomponent of working memory in mental arithmetic might be subject to linguistic, instructional, and contextual factors.

Three Ehrhart quasi-polynomials

Let 𝔭(b)⊂ℝ d be a semi-rational parametric polytope, where b=(b j )∈ℝ N is a real multi-parameter. We study intermediate sums of polynomial functions h(x) on 𝔭(b),

Probabilistic Aggregation of Uncertain Geological Resources

Commodities such as oil and gas occur in isolated reservoirs or accumulations, more generically called basic units here. To understand a study area’s economic potential and to craft plans for exploration and development, resource analysts often aggregate (sum, accumulate) basic unit magnitudes in distinct spatial subsets of the study area and then appraise the total area’s potential by summing these intermediate sums. In a probabilistic approach, magnitudes are modeled as random variables. Some have asked, “Do different methods of partitioning basic units into subsets lead to different probability distributions for the sum of all basic unit magnitudes?” Any method of aggregation of basic unit magnitudes which obeys the rules of probability leads to the same probability distribution of the sum of all unit magnitudes as that computed by direct summation of all basic unit magnitudes. A Monte Carlo simulation of a synthetic example in which the magnitude of resource in each unit is marginally lognormal and pairwise correlations among basic unit magnitudes are specified illustrates key features of probabilistic aggregation. The joint distribution of certain pairs of aggregates are closely approximated by a bivariate lognormal distribution.

INTERMEDIATE SUMS ON POLYHEDRA II: BIDEGREE AND POISSON FORMULA

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these intermediate sums by means of the intermediate generating functions $S^L(s+c)(\xi)$, and expose the bidegree structure in parameters s and $\xi$, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra, Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: Computation and real Ehrhart theory, Mathematika 59 (2013), 1-22]. The bidegree structure is key to a new proof for the Baldoni--Berline--Vergne approximation theorem for discrete generating functions [Local Euler--Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes, Contemp. Math. 452 (2008), 15-33], using the Fourier analysis with respect to the parameter s and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

Activation of the intermediate sum in intentional and automatic calculations.

Most research investigating how the cognitive system deals with arithmetic has focused on the processing of two addends. Arithmetic that involves more addends has specific cognitive demands such as the need to compute and hold the intermediate sum. This study examines the intermediate sums activations in intentional and automatic calculations. Experiment 1 included addition problems containing three operands. Participants were asked to calculate the sum and to remember the digits that appeared in the problem. The results revealed an interference effect in which it was hard to identify that the digit representing the intermediate sum was not actually one of the operands. Experiment 2, further examined if the intermediate sum is activated automatically when a task does not require calculation. Here, participants were presented with a prime of an addition problem followed by a target number. The task was to determine whether the target number is odd or even, while ignoring the addition problem in the prime. The results suggested that the intermediate sum of the addition problem in the prime was activated automatically and facilitated the target. Overall, the implications of those findings in the context of theories that relate to cognitive mathematical calculation is further discussed.

Fermion pairing and the scalar boson of the 2D conformal anomaly

We analyze the phenomenon of fermion pairing into an effective boson associated with anomalies and the anomalous commutators of currents, bilinear in the fermion fields. In two spacetime dimensions the chiral bosonization of the Schwinger model is determined by the chiral current anomaly of massless Dirac fermions. A similar bosonized description applies to the 2D conformal trace anomaly of the fermion stress-energy tensor. For both the chiral and conformal anomalies, correlation functions involving anomalous currents, j 5 or T μν of massless fermions exhibit a massless boson 1/k 2 pole, and the associated spectral functions obey a UV finite sum rule, becoming δ-functions in the massless limit. In both cases the corresponding effective action of the anomaly is non-local, but may be expressed in a local form by the introduction of a new bosonic field, which becomes a bona fide propagating quantum field in its own right. In both cases this is expressed in Fock space by the anomalous Schwinger commutators of currents becoming the canonical commutation relations of the corresponding boson. The boson has a Fock space operator realization as a coherent superposition of massless fermion pairs, which saturates the intermediate state sums in quantum correlation functions of fermion currents. The Casimir energy of fermions on a finite spatial interval [0, L] can also be described as a coherent scalar condensation of pairs, and the one-loop correlation function of any number n of fermion stress-energy tensors 〈TT . . . T 〉 may be expressed as a combinatoric sum of n!/2 linear tree diagrams of the scalar boson.

INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

Young Gauss Meets Dynamical Systems

But we know that one who does this by merely adding terms one after another is not seeing the forest for the trees. An anecdote about young Gauss tells us that he solved the above problem by noticing that pairwise addition of terms from opposite ends of the list yields identical intermediate sums. This famous story is well told by Hayes in [5], with references. A very convenient way to express Gauss’s idea is to write down the series twice, once in ascending and once in descending order,

Response of temperate grasslands at different altitudes to simulated summer drought differed but scaled with annual precipitation

Abstract. Water is an important resource for plant life. Since climate scenarios for Switzerland predict an average reduction of 20% in summer precipitation until 2070, understanding ecosystem responses to water shortage, e.g. in terms of plant productivity, is of major concern. Thus, we tested the effects of simulated summer drought on three managed grasslands along an altitudinal gradient in Switzerland from 2005 to 2007, representing typical management intensities at the respective altitude. We assessed the effects of experimental drought on above- and below-ground productivity, stand structure (LAI and vegetation height) and resource use (carbon and water). Responses of community above-ground productivity to reduced precipitation input differed among the three sites but scaled positively with total annual precipitation at the sites (R2=0.85). Annual community above-ground biomass productivity was significantly reduced by summer drought at the alpine site receiving the least amount of annual precipitation, while no significant decrease (rather an increase) was observed at the pre-alpine site receiving highest precipitation amounts in all three years. At the lowland site (intermediate precipitation sums), biomass productivity significantly decreased in response to drought only in the third year, after showing increased abundance of a drought tolerant weed species in the second year. No significant change in below-ground biomass productivity was observed at any of the sites in response to simulated summer drought. However, vegetation carbon isotope ratios increased under drought conditions, indicating an increase in water use efficiency. We conclude that there is no general drought response of Swiss grasslands, but that sites with lower annual precipitation seem to be more vulnerable to summer drought than sites with higher annual precipitation, and thus site-specific adaptation of management strategies will be needed, especially in regions with low annual precipitation.

RADIX MODULAR MULTIPLICATION ALGORITHM

In this paper, the concept of a new Radix Modular Multiplication Algorithm (MMA) is proposed. The novelty of the new Radix-2n MMA is that the intermediate partial sums (IPSs) are not restricted to be less than the modulus M, but only to be represented by N bits, where N is the number of bits needed to represent the modulus M. Hence, the IPSs become redundant to the modulus M. Two new Radix-2n MMAs (for n=2 and 4) based on the proposed concept are considered in detail as well. It is shown that a parallel multiplier based on the new Radix-4 MMA achieves twice the speed of a parallel multiplier based on a recent Radix-2 MMA. This result becomes more significant when it is noted that doubling the speed was achieved without any increase in the hardware requirement. In addition, it is shown that the parallel multiplier based on the new Radix-16 MMA achieves four times the speed of that of the Radix-2 MMA with the same hardware requirement. When compared to the existing Radix-4 MMAs that are based on Carry Save format and Binary Signed Digit (BSD) representation, it is shown that the delay per step of the proposed Radix-4 multiplier is decreased by more than 40% when the intermediate steps are implemented using Carry Save Adders (CSAs).

Intermediate sums and stochastic compactness of maxima

A connection between the convergence in distribution of sums of intermediate order statistics and the stochastic compactness of maxima is established.

The Asymptotic Distribution of Intermediate Sums

Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive numbers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, and consider the sums $I_n(a, b) = \sum^{\lbrack bk_n\rbrack}_{i=\lbrack ak_n\rbrack+1} X_{n+1-i,n}$ of intermediate order statistics, where $0 < a < b$. We find necessary and sufficient conditions for the existence of constants $A_n > 0$ and $C_n$ such that $A^{-1}_n(I_n(a,b) - C_n)$ converges in distribution along subsequences of the positive integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions.

Multiphoton-detachment cross sections for H-

Perturbation theory is applied to the evaluation of the multiphoton-detachment cross sections of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ in a number of approximations. The zero-range plane-wave approximation is the simplest, as its effective dipole matrix element is reducible to a recursive differentiation, and it is applied to the one- to seven-photon-detachment processes. Improvements on this are made in the bound state by means of a short-range potential model and a dynamic screening representation, and in the continuum wave functions with the use of the best previously calculated e-H(1s) singlet scattering phase shifts. The dynamic screening parameters are adjusted to give a good fit of the resulting one-photon cross section to its correct well-known value, and the accurate phase shifts effectively include all the important correlation effects in the free-free amplitudes. These improved wave functions are used for the evaluation of the two- and three-photon generalized detachment cross sections by explicitly carrying out all intermediate state sums. Our best \ensuremath{\sigma}${\mathrm{^}}_{2}$ and \ensuremath{\sigma}${\mathrm{^}}_{3}$ are estimated to be accurate to order 5%. The present theoretical results are compared with other calculations and recent measurements, revealing areas of agreement as well as certain discrepancies.

Best “ordering” for floating-point addition

This correspondence examines the influence of the summation order on the numerical accuracy of the resulting sum when the addition is performed using floating-point, finite-precision arithmetic. A simple statistical model is used to find the relative performance of various addition procedures. It is shown that phrasing the question in terms of finding a best ordering is overly restrictive since the most natural and accurate procedures for performing the addition utilize the storage of intermediate sums as well as performing orderings.

Convergence behaviour in chiral bag models and the pion-nucleon scattering amplitude

In typical chiral-bag-model calculations, strong restrictions are imposed on the quark configurations contributing to intermediate-state sums without addressing the question of convergence. In this paper the convergence properties of the sums are examined in connection with the pion-nucleon amplitude evaluated perturbatively to lowest order. Two forms of the interaction are employed: one where the pion-quark coupling is confined to the bag surface and one where it extends throughout the bag volume. The convergence behaviour with respect to intermediate sums is quite different in the two models. Our numerical results for the πN amplitude using the surface interaction exhibit rather poor convergence properties, with large cancellations occurring between intermediate particle and antiparticle contributions. The convergence behaviour obtained with the volume interaction is substantially better. Some implications of our results for future chiral-model calculations are presented.

Quantum fluctuations in the bag and nucleon observables

A time-dependent, perturbative formalism is developed to describe the effect of quantum fluctuations of the quark fields inside a hadronic bag on physical observables. This formalism differs from previous ones in that (i) all contributions to second order in the QCD coupling constant are generated systematically and included in the numerical treatment; (ii) the intermediate sums are carried out over a full set of states consistent with parity and SU(2) coupling rules. After removing the logarithmic divergences arising from the intermediate state sums in the 1-body terms and applying a finite renormalization, the second order fluctuation corrections to several proton and neutron observables are computed. The results obtained are significantly different from those obtained in earlier calculations. In no case considered are the calculated corrections able to account for existing discrepancies between empirical values and lowest order hadronic bag results for nucleon observables. The results obtained for the magnetic moments indicate a possible difficulty in the renormalization prescription when applied in the static, non-Lorentz invariant approximation to the bag model.

Bilocal form factors and Regge couplings

The infinitely many local observables defined by expanding the bilocal currents j μ ( x, y) on the light cone in powers of ( x− y) μ are used to study scattering processes where one or two external lines are Reggeons. The algebra of bilocal form factors of Fritzsch and Gell-Mann implies the existenc of an algebra of infinitely many form factors F J± ( bdk) of any spin J, definite signature±, and arbitrary momentum transfer k. These “signatured form factors“ can be continued analytically in k as well as in J and superconvergence relations are obtained for the couplings of strongly interacting particles of arbitrarily high spin and for Regge couplings. Also the commutator of form factors with Regge residues vanishes, except at certain momentum transfers. In particular, the charges F Jη (0) of spin J and signature η = (−) J act as “daughter lowering operators”. The range of validity of these sum rules in momentum transfer is discussed by continuing the scattering amplitudes involving the spin J and spin J′ currents analytically in J and J′ When the sum rules break down one can truncate the intermediate sums at some value of the intermediate masses. In this way one obtains an algebraic form of finite-energy sum rules. They consist in commutation rules among Reggeons with the right-hand side being given by the Regge couplings that can be exchanged in the corresponding scattering process. The structure constants of this algebra are the corresponding triple-Regge couplings. As an example one may assume dominance of leading trajectories and finds that ϱ, A 1, π and f residues commute like O(5). Including also the commutators with vector and axial vector changes, one obtains the “supergroup” SU(2) × SU(2) × O(5). The Regge couplings we are dealing with here are all in the infinite-momentum frame. In order to give the connection with standard t-channel couplings the angular condition for bilocal form factors are derived and continued in angular momentum. As another technical side results we point out that under very weak assumptions the bilocal form factors F( k , z) are analytic in z. F( k , z) does not, however, possess Regge behaviour for | z| → ∞ as often stated. Only signatured bilocal form factors do, as shown in this work.

The algebra of currents as a complete dynamical method in the nuclear many-body problem: Application to an exactly soluble model

A method for obtaining approximate solutions to the standard shell-model problem, as applied to collective states, is described which exploits two features of such problems. One is the completely general mathematical feature that the Hamiltonian is a simple polynomial in the generators of a Lie group. The second is the widespread physical feature that when the matrix element of a product of generators, taken between suitable low-lying states, is evaluated by sum rule techniques, the intermediate state sums are exhausted by a few terms (collective states). Such sum rule techniques can be applied to the Lie algebra, the Casimir operators of the algebra (whose eigenvalues specify the representations appropriate to the fermion system under study), to the condition that the Hamiltonian be diagonal in the space of its eigenstates, and to the equations of motion for the generators taken between such states. In this paper the technique is applied to an exactly soluble model introduced by Meshkov, Lipkin, and Glick. It is seen that the sum rules can be cut at various levels of approximation and with sufficient care in the way this is done that the resulting systems of non-linear equations among physical matrix elements constitutes a complete (or even overcomplete but reasonably self-consistent) dynamical scheme for the computation of a specified set of physical quantities. Even the crudest such approximation gives qualitatively reasonable results right through the phase transition exhibited by the model. The best approximation carried out yields all the physical quantities associated with the lowest four eigenstates of the system, i.e. up to three phonon-states in the vibrational regime. Essentially exact results for all coupling constants are obtained for the quantities normally calculated by the random phase approximation and good values for the remaining quantities. The strength of the method is seen to be that for systems with collective modes, the complexity of its application depends only on the structure of the group studied and on the number of physical states taken into account and is independent of the total number of particles treated.

Hard-Pion Current-Algebra Calculation of Meson Processes—N-Point Functions

Hard-pion techniques are presented for calculating $T$ products of an arbitrary number of vector and axial-vector currents under the assumptions of single $\ensuremath{\sigma}$-, $\ensuremath{\pi}$-, $\ensuremath{\rho}$-, and ${A}_{1}$-meson saturation of intermediate sums, chiral $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$-algebra commutation relations, conservation of vector current (CVC), and partial conservation of axial-vector current (PCAC). The single-meson dominance hypothesis is shown to imply that one calculates the $T$ products, keeping only certain generalized tree and seagull diagrams. Alternatively, the assumption can be replaced by requiring that one calculate with an "effective" interaction Lagrangian ${\mathcal{L}}_{I}$ to lowest nonvanishing order. The conditions that the remaining hypotheses (current commutation relations, CVC, and PCAC) impose on ${\mathcal{L}}_{I}$ are expressed in terms of functional differential equations to determine the form of ${\mathcal{L}}_{I}$. These equations are shown to be consistent with each other and may in fact be integrated order by order. The ${\mathcal{L}}_{I}$ needed to calculate any four-point function is given explicitly.