Explicit Expansion Research Articles

Can working memory capacity be expanded by boosting working memory updating efficiency in older adults?

Working memory updating (updating) and working memory capacity (WMC) have been assumed to share a common mechanism. However, it is unclear whether WMC can be expanded by boosting the efficiency of updating, particularly during late adulthood. In this randomized controlled study, 33 older adults (aged 60 years and above, M = 69.53, SD = 5.21) were assigned to updating training (n = 17) and contact control (n = 16) groups. In the training group, updating was targeted by a running memory task and a chess game in each training session; whereas in the control group, motivational effects were estimated by their attendance to a series of mental health-related lectures. The boosted updating efficiency was demonstrated by direct training gains over sessions on updating criterion/training tasks, and significant frontocentral inflation at the N2 time range. Near transfer effects were revealed on untrained similar measures immediately after training and partly maintained at a 3-month follow-up. Far transfer effects to WMC were revealed by a significant increase in contralateral delay activity (CDA) ERP measures, although these effects did not manifest on WMC span tasks and other far transfer measures. Note that increased CDA was significantly correlated with the extent of direct gains in updating training. Our findings suggest that updating efficiency can be boosted in older adults; although it has not yet brought about an explicit expansion of WMC in our behavioral measures (i.e., on span tasks), capacity-related neural changes (i.e., on CDA) have nevertheless, started to show up in our short-term updating training program. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

A 3D parallel boundary element method on unstructured triangular grids for fully nonlinear wave-body interactions

This paper presents the development and validation of a three-dimensional numerical wave tank devoted to studying wave-structure interaction problems. It is based on the fully nonlinear potential flow theory, here solved by a boundary element approach and using unstructured triangular meshes of the domain's boundaries. Time updating is based on a second-order explicit Taylor series expansion. The method is parallelized using the Message Passing Interface (MPI) in order to take advantage of multi-processor systems. For radiation problems, with cylindrical bodies moving in prescribed motion, the free-surface is updated with a fully Lagrangian scheme, and is able to reproduce reference results for nonlinear forces exerted on the moving body. For diffraction problems, semi-Lagrangian time-updating is used, and reproduces nonlinear effects for diffraction on monopiles. Finally, we study the nonlinear wave loads on a fixed semi-submersible structure, thereby illustrating the possibility to apply the proposed numerical model for the design of offshore structures and floaters.

Recovering Quantum Gates from Few Average Gate Fidelities.

Characterizing quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today's devices. One method for characterizing processes is randomized benchmarking, which is robust against state preparation and measurement errors and can be used to benchmark Clifford gates. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. In this Letter, we show that the favorable features of both approaches can be combined. For characterizing multiqubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured with respect to random Clifford unitaries. Moreover, for general unital quantum channels, we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity-a figure of merit characterizing the coherence of a process.

CDS calibration under an extended JDCEV model

ABSTRACTWe propose a new methodology for the calibration of a hybrid credit-equity model to credit default swap (CDS) spreads and survival probabilities. We consider an extended Jump to Default Constant Elasticity of Variance model incorporating stochastic and possibly negative interest rates. Our approach is based on a perturbation technique that provides an explicit asymptotic expansion of the CDS spreads. The robustness and efficiency of the method is confirmed by several calibration tests on real market data.

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer m(m≥1), we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for m=2, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.

Simulating the game-theoretic market equilibrium and contract-driven investment in global gas trade using an agent-based method

To understand how the alternative US liquefied natural gas exportation strategies may affect future global gas market dynamics, a global-scale model Gas-GAME is developed using an agent-based framework. This is the first model having explicit contract-driven capacity expansion process, allowing investors to hold imperfect foresights, and simulating market power in global gas trade. With these features, Gas-GAME can analyse market development subject to the incentives and perspectives of each market player. The model simulates short-term game-theoretical market equilibrium with Mixed Complementarity Problem approach. For long-term investment decisions, bilateral contracting processes considering both import requests and export profitability are modelled. A base case is presented and validated, followed by a case study considering US export strategy. When the US stays conservative in export expansion, gas supply tightness occurs, leading to continuing European dependence on Russian gas, and a shift to pipeline-based import in the Chinese market. Conversely, when the US invests aggressively, Middle East and Australia both see significant revenue losses, and Western Europe constructs more regasification plants to provide alternatives to Russian supply. Gas-GAME captures the essential dynamics between market power, short-term prices and long-term contracts to provide a more nuanced view of global gas market.

From Partition Identities to a Combinatorial Approach to Explicit Satake Inversion

In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands’ beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of $${{\rm Sym}^j}$$ $${{{\rm Sym}^{k}}V}$$ for $${{\rm GL}_2}$$ in the case k = 3. We compute the plethysm explicitly for the cases k = 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of $${{\rm GL}_2}$$ for these two cases.

Bridging the gap between constant step size stochastic gradient descent and Markov chains

We consider the minimization of a strongly convex objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step size, as well as the lack of convergence in the general (nonquadratic) case. For this analysis we bring tools from Markov chain theory into the analysis of stochastic gradient. We then show that Richardson–Romberg extrapolation may be used to get closer to the global optimum, and we show empirical improvements of the new extrapolation scheme.

The contribution of scalars to [formula omitted] SYM amplitudes II: Young tableaux, asymptotic factorisation and strong coupling

We disentangle the contribution of scalars to the OPE series of null hexagonal Wilson loops/MHV gluon scattering amplitudes in multicolour N=4 SYM. In specific, we develop a systematic computation of the SU(4) matrix part of the Wilson loop by means of Young tableaux (with several examples). Then, we use a peculiar factorisation property (when a group of rapidities becomes large) to deduce an explicit polar form. Furthermore, we emphasise the advantages of expanding the logarithm of the Wilson loop in terms of ‘connected functions’ as we apply this procedure to find an explicit strong coupling expansion (definitively proving that the leading order can prevail on the classical AdS5 string contribution).

Characterizing the number of coloured [formula omitted]-ary partitions modulo [formula omitted], with and without gaps

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of m-ary partitions modulo m, with and without gaps. In this paper we extend these results to the case of coloured m-ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo m.

The Asymptotic of Transmission Eigenvalues for a Domain with a Thin Coating

We consider the transmission eigenvalue problem for a medium surrounded by a thin layer of inhomogeneous material with different refractive index. We derive explicit asymptotic expansion for the transmission eigenvalues with respect to the thickness of the thin layer. We prove the error estimate for the asymptotic expansion up to order 1. This expansion can be used to obtain explicit expressions for a constant index of refraction.

An expansion in the model space in the context of utility maximization

In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.

Higher weight on GL(3). I: The Eisenstein series

AbstractThe purpose of this paper is to collect and make explicit the results of Langlands [16], Bump [3], Miyazaki [18] and Manabe, Ishii and Oda [17] for theGL⁢(3){\mathrm{GL}(3)}Eisenstein series and Whittaker functions which are non-trivial onSO⁢(3,ℝ){\mathrm{SO}(3,\mathbb{R})}. The final goal for the series of papers is a complete and completely explicit spectral expansion forL2⁢(SL⁢(3,ℤ)∖SL⁢(3,ℝ)){L^{2}(\mathrm{SL}(3,\mathbb{Z})\setminus\mathrm{SL}(3,\mathbb{R}))}in the style of Duke, Friedlander and Iwaniec’s paper [8]. We derive a number of new results on the Whittaker functions and Eisenstein series, and give new, concrete proofs of the functional equations and spectral expansion in place of the general constructions of Langlands.

Construction and implementation of asymptotic expansions for Laguerre-type orthogonal polynomials

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical Laguerre polynomials correspond to $Q(x)=x$. The computation of higher-order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full description seems to be lacking in literature. However, this information is implicitly available in the work of Vanlessen, based on a non-linear steepest descent analysis of an associated so-called Riemann--Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higher-order terms in the asymptotic expansions of Laguerre and Laguerre-type polynomials. This effort is similar to the case of Jacobi and Jacobi-type polynomials in a previous paper. We supply an implementation with explicit expansions in four different regions of the complex plane. These expansions can also be extended to Hermite-type weights of the form $\exp(-\sum_{k=0}^m q_k x^{2k})$ on $(-\infty,\infty)$, and to general non-polynomial functions $Q(x)$ using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules in a lower computational complexity than based on the recurrence relation, and with improved accuracy for large degree. They are also of interest in random matrix theory.

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

We study a dyadic branching Brownian motion on the real line with absorption at 0, drift μ∈R and started from a single particle at position x>0. With K(∞) the (possibly infinite) total number of individuals absorbed at 0 over all time, we consider the functions ωs(x):=Ex[sK(∞)] for s≥0. In the regime where μ is large enough so that K(∞)<∞ almost surely and that the process has a positive probability of survival, we show that ωs<∞ if and only if s∈[0,s0] for some s0>1 and we study the properties of these functions. Furthermore, ω(x):=ω0(x)=Px(K(∞)=0) is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption.We give descriptions of the family ωs,s∈[0,s0] through the single pair of functions ω0(x) and ωs0(x), as extremal solutions of the Kolmogorov–Petrovskii–Piskunov (KPP) travelling wave equation on the half-line, through a martingale representation, and as a single explicit series expansion. We also obtain a precise result concerning the tail behaviour of K(∞). In addition, in the regime where K(∞)>0 almost surely, we show that u(x,t):=Px(K(t)=0) suitably centred converges to the KPP critical travelling wave on the whole real line.

On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

Analysis of Underactuated Dynamic Locomotion Systems Using Perturbation Expansion: The Twistcar Toy Example

Underactuated robotic locomotion systems are commonly represented by nonholonomic constraints where in mixed systems, these constraints are also combined with momentum evolution equations. Such systems have been analyzed in the literature by exploiting symmetries and utilizing advanced geometric methods. These works typically assume that the shape variables are directly controlled, and obtain the system’s solutions only via numerical integration. In this work, we demonstrate utilization of the perturbation expansion method for analyzing a model example of mixed locomotion system—the twistcar toy vehicle, which is a variant of the well-studied roller-racer model. The system is investigated by assuming small-amplitude oscillatory inputs of either steering angle (kinematic) or steering torque (mechanical), and explicit expansions for the system’s solutions under both types of actuation are obtained. These expressions enable analyzing the dependence of the system’s dynamic behavior on the vehicle’s structural parameters and actuation type. In particular, we study the reversal in direction of motion under steering angle oscillations about the unfolded configuration, as well as influence of the choice of actuation type on convergence properties of the motion. Some of the findings are demonstrated qualitatively by reporting preliminary motion experiments with a modular robotic prototype of the vehicle.

Variants of the RSK algorithm adapted to combinatorial Macdonald polynomials

We introduce variations of the Robinson–Schensted correspondence parametrized by positive integers p. Each variation gives a bijection between permutations and pairs of standard tableaux of the same shape. In addition to sharing many of the properties of the classical Schensted algorithm, the new algorithms are designed to be compatible with certain permutation statistics introduced by Haglund in the study of Macdonald polynomials. In particular, these algorithms provide an elementary bijective proof converting Haglund's combinatorial formula for Macdonald polynomials to an explicit combinatorial Schur expansion of Macdonald polynomials indexed by partitions μ satisfying μ1≤3 and μ2≤2. We challenge the research community to extend this RSK-based approach to more general classes of partitions.

Modular embeddings of Teichmüller curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

On solving an isospectral flow

In this paper we expand the solution of the matrix ordinary differential system, originally due to Bloch and Iserles, of the form X′=[N,X2],t≥0,X(0)=X0∈Sym(n),N∈so(n), where Sym(n) denotes the space of real n×n symmetric matrices and so(n) denotes the Lie algebra of real n×n skew-symmetric matrices. The flow is solved using explicit Magnus expansion, which respects the isospectrality of the system. We represent the terms of expansion as binary rooted trees and deduce an explicit formalism to construct the trees recursively.