Diagonal Perturbations Research Articles

Pelvis perturbations in various directions while standing in staggered stance elicit concurrent responses in both the sagittal and frontal plane.

Increasing knowledge on human balance recovery strategies is important for the development of balance assistance strategies using assistive devices like a powered lower-limb exoskeleton. One of the postures which is relevant for this scenario, but underexposed in research, is staggered stance, a posture with one foot in front. We therefore aimed to gain a better understanding of balance recovery in staggered stance. We studied balance responses at joint- and muscle levels to pelvis perturbations in various directions while standing in this posture. Ten healthy individuals participated in this study. We used one actuator beside and one behind the participant to apply 150 ms perturbations in mediolateral (ML), anteroposterior (AP) and diagonal directions, with a magnitude of 3, 6, 9 and 12% of the participant's body weight (BW). Meanwhile, motion capture, ground reaction forces and moments, and electromyography of the muscles around the ankles and hips were recorded. The perturbations caused movements of the centre of mass (CoM) and centre of pressure (CoP) in the direction of the perturbation. These were often accompanied by motions in a direction different from the perturbation direction. After perturbations perpendicular to the line between both feet, large and significant AP deviations were present of the CoM (-0.27 till 0.40 cm/%BW, p < 0.029) and CoP (-0.99 till 0.80 cm/%BW, p < 0.001). Also, stronger responses on joint and muscle level were present after these perturbations, compared to AP and diagonal perturbations collinear with the line between both feet. The hip, knee and ankle joints contributed differently to the balance responses after the different perturbation directions. To conclude, standing in a staggered stance posture makes individuals more vulnerable to perturbations perpendicular to the line between both feet, requiring larger responses on joint level as well as contributions in the sagittal plane.

Robustness versus sensitivity in non-Hermitian topological lattices probed by pseudospectra

Non-Hermitian topological systems simultaneously possess two antagonistic features: ultrasensitivity due to exceptional points and robustness of topological zero-energy modes, and it is unclear which one prevails under different perturbations. We study that question by applying the pseudospectrum theory on the prototypical non-Hermitian Su-Schrieffer-Heeger lattice. Topological modes around the underlying third-order exceptional point (EP3) are robust with respect to chiral perturbations but sensitive to diagonal perturbations. In fact, exactly at the EP3 the chiral symmetry leads to a suppressed sensitivity, that corresponds to an EP2. Finally, for nonlinearly induced perturbations we provide a connection between the pseudospectrum approach and a nonlinear phase shift, which is relevant for experiments.

System Monotonicity and Subspace Tracking: A Geometric Perspective of the Frisch–Shapiro Scheme

The Shapiro scheme, together with the closely related Frisch–Kalman scheme, has been an important approach to system identification and statistical analysis. A longstanding result on this scheme, known as the Shapiro theorem, is both informative and significant. This article imparts a geometric understanding to the Shapiro theorem and generalizes it to the asymmetric setting using the notion of cone-invariance. In particular, we establish the equivalence between two important properties of a real-valued square matrix—irreducible orthant-invariance and simplicity of its dominant eigenvalue under arbitrary diagonal perturbations. The result can be regarded as a converse Perron–Frobenius theorem. Furthermore, we investigate two applications of the proposed result in systems and control, namely, characterization of irreducibly orthant-monotone nonlinear systems and subspace tracking via decentralized control. We also extend the established result to accommodating polyhedral cones and obtain several insights.

Acute Effects of a Perturbation-Based Balance Training on Cognitive Performance in Healthy Older Adults: A Pilot Study.

Aging is accompanied by an alteration in the capacity to ambulate, react to external balance perturbations, and resolve cognitive tasks. Perturbation-based balance training has been used to induce adaptations of gait stability and reduce fall risk. The compensatory reactions generated in response to external perturbations depend on the activation of specific neural structures. This suggests that training balance recovery reactions should show acute cognitive training effects. This study aims to investigate whether exposure to repeated balance perturbations while walking can produce acute aftereffects that improve proactive and reactive strategies to control gait stability and cognitive performance in healthy older adults. It is expected that an adaptation of the recovery reactions would be associated with increased selective attention and information processing speed. Twenty-eight healthy older adults were assigned to either an Experimental (EG) or a Control Group (CG). The protocol was divided in 2 days. During the first visit, all participants completed the Symbol Digit Modalities Test (SDMT) and the Trail Making Test (TMT). During the second visit, a cable-driven robot was used to apply waist-pull perturbations while walking on a treadmill. The EG was trained with multidirectional perturbations of increasing intensity. The CG walked for a comparable amount of time with cables on, but without experiencing perturbations. Before and after the training, all participants were exposed to diagonal waist-pull perturbations. Changes in gait stability were evaluated by comparing the distance between the heel of the leading leg and the extrapolated Center of Mass (Heel-XCoM Distance—HXD) at perturbation onset (PON) and first compensatory heel strike (CHS). Finally, the cables were removed, and participants completed the SDMT and the TMT again. Results showed that only the EG adapted the gait stability (p < 0.001) in reaction to diagonal perturbations and showed improved performance in the SDMT (p < 0.001). This study provides the first evidence that a single session of perturbation-based balance training produce acute aftereffects in terms of increased cognitive performance and gait stability in healthy older adults. Future studies will include measures of functional activation of the cerebral cortex and examine whether a multi-session training will demonstrate chronic effects.

Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

Emergence of synchronous behaviors for the Schrödinger–Lohe model with frustration

We introduce the Schrödinger–Lohe model (S–L) with interaction frustration which is a toy model for quantum synchronization, and study its emergent collective dynamics. Our proposed model might not capture quantum entanglement which is one of genuine quantum phenomenon as it is, however it exhibits collective synchronous behaviors and can be reduced to the Kuramoto model with frustration in the case of spatial homogeneity and unit mass. For the emergent dynamics of the model, we present two sufficient frameworks leading to the complete and practical synchronizations for identical and non-identical oscillators, respectively, in terms of frustration function, system parameters and initial data. When the frustration functions admit real diagonal perturbations, complete synchronization emerges for identical ensemble with some class of initial data, in contrast, for non-identical ensemble, we can expect practical synchronization in which the diameter for wave functions can be controlled by tuning the coupling strength. For purely imaginary valued frustrations, we analytically show that complete or practical synchronization cannot occur for generic initial data. This illustrates that the S–L model exhibits some interesting dynamic patterns different from the Kuramoto synchronization. We provide several numerical examples which manifest periodic behaviors, reminiscent of the Kuramoto model with cosine coupling.

Spectral edge behavior for eventually monotone Jacobi and Verblunsky coefficients

We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic behavior of the spectral density at the top of the essential spectrum. In particular, allowing on both diagonal and off-diagonal Jacobi parameters perturbations of the free case of the form - \sum_{j=1}^J c_j n^{-\tau_j} + o(n^{-\tau_1-1}) with 0 &lt; \tau_1 &lt; \tau_2 &lt; \dots &lt; \tau_J and c_1 &gt; 0 , we find the asymptotic behavior of the log of spectral density to order O(\mathrm {log}(2-x)) as x approaches 2. Apart from its intrinsic interest, the above results also allow us to describe the asymptotics of the spectral density for orthogonal polynomials on the unit circle with real-valued Verblunsky coefficients of the same form.

Qualification Conditions in Semialgebraic Programming

For an arbitrary finite family of semialgebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies the Mangasarian--Fromovitz constraint qualification. Using the Milnor--Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of “regular” problems approximating an arbitrary semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.

A single session of perturbation-based gait training with the A-TPAD improves dynamic stability in healthy young subjects.

Gait and balance disorders are among the most common causes of falls in older adults. Most falls occur as a result of unexpected hazards while walking. In order to improve the effectiveness of current fall-prevention programs, new balance training paradigms aim to strengthen the control of the compensatory responses required after external perturbations. The aim of this study was to analyze the adaptions of reactive and proactive strategies to control stability after repeated exposures to waist-pull perturbations delivered while walking. Eight healthy young subjects participated in a single training session with the Active Tethered Pelvic assisted Device (A-TPAD). Participants were exposed to repeated multi-directional perturbations of increasing intensity. The Antero-Posterior (AP) and Medio-Lateral (ML) Base of Support (BoS) and Margin of Stability (MoS) during the response to diagonal perturbations were compared before and after the training. Results showed that participants adapted both the reactive and proactive strategies to control walking balance by significantly increasing their pre- and post-perturbation stability. The changes were principally accounted for by an increment of the AP BoS and MoS and a reduction of ML BoS. This improved their ability to react to a diagonal perturbation. We envision that this system can be used to develop a perturbation-based gait training aimed at improving balance and control of stability during walking, thus reducing fall risk.

Sign-changing diagonal perturbations of Laplacian matrices of graphs

In this paper we provide sufficient conditions for positive (semi)definiteness of sign-changing diagonal perturbations of positive semidefinite difference operators and their matrix representations, the Laplacian matrices of graphs. Our estimates arise from the discrete version of the Poincaré inequality and essentially depend on the algebraic connectivity of the underlying graph, i.e., the second smallest eigenvalue of the graph Laplacian matrix. We generalize our results to positive semidefinite matrices with simple zero eigenvalue and illustrate our results by numerical experiments and discuss the optimality of our assumptions.

35-GHz Wideband Circularly Polarized Patch Array on LTCC

A 35-GHz wideband circularly polarized (CP) patch array of $2\times 2$ elements using low-temperature cofired ceramic technology is presented. The widened 3-dB axial ratio (AR) bandwidth is achieved by constructing a $1 \times 2$ sequentially rotated CP patch array with small inner-element spacing in the antenna element. The antenna element consists of two close circular patches with diagonal perturbations, and the patches are rotated with respect to each other. The two patches are fed with equal amplitude and orthogonal phase difference using one probe, which has one vertical part and two different horizontal parts. The fabricated array with a dimension of $15.3 \times 28.3\times 1.3$ mm3 shows wide −10-dB $\mid S_{11} \mid $ and 3-dB AR bandwidths of over 29.6% and 26%, respectively. The measured maximum gain of the four-element array is 9.3 dBi with slight fluctuations over 31–39 GHz frequency range.

Quasi-periodic oscillations from relativistic ray-traced hydrodynamical tori

We simulate an oscillating purely hydrodynamical torus with constant specific angular mo- mentum around a Schwarzschild black hole. The goal is to search for quasi-periodic oscil- lations (QPOs) in the light curve of the torus. The initial torus setup is subjected to radial, vertical and diagonal (combination of radial and vertical) velocity perturbations. The hydro- dynamical simulations are performed using the general relativistic magnetohydrodynamics code Cosmos++ and ray-traced using the GYOTO code. We found that a horizontal velocity perturbation triggers the radial and plus modes, while a vertical velocity perturbation trig- gers the vertical and X modes. The diagonal perturbation gives a combination of the modes triggered in the radial and vertical perturbations.

Zero Assignment Problem in RLC Networks

The paper deals with the problem of zero assignment in RLC networks by selection of appropriate values for the non- dynamical elements, the resistors. For a certain family of network redesign problems by the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem belongs to the family of DAP problems (Determinantal Assignment Problem) and has common features with the pole assignment problem by decentralized output feedback and the zero assignment problem via structured additive perturbations. We demonstrate that the sufficient condition for generic zero assignment by selecting the resistors holds true. This condition is related to the rank of the differential of the related map and holds true generically when the degrees of freedom of the matrix of resistors exceeds the number of frequencies to be assigned (n > p + q). Using this result, we show through a generic example that the sufficient condition for the general zero assignment problems in RLC networks is satisfied and thus, zero assignment can be achieved via resistor determination.

Relaxing Nonconvex Quadratic Functions by Multiple Adaptive Diagonal Perturbations

The current bottleneck of globally solving mixed-integer (non-convex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our resulting relaxation does not use significantly more variables than the original problem, in contrast to many other relaxations based on lifting. The corresponding separation problem is a highly structured semidefinite program (SDP) with convex but non-smooth objective. We propose to solve this separation problem with a specialized primal-barrier coordinate minimization algorithm. Computational results show that our approach is very promising. First, our separation algorithm is at least an order of magnitude faster than interior point methods for SDPs on problems up to a few hundred variables. Secondly, on nonconvex quadratic integer problems, our cutting surface procedure provides lower bounds of almost the same strength with the diagonal SDP bounds used by (Buchheim and Wiegele, 2013) in their branch-and-bound code Q-MIST, while our procedure is at least an order of magnitude faster on problems with dimension greater than 70. Finally, combined with (linear) projected RLT cutting planes proposed by (Saxena, Bonami and Lee, 2011), our procedure provides slightly weaker bounds than their projected SDP+RLT cutting surface procedure, but in several order of magnitude shorter time. Finally we discuss various avenues to extend our work to design more efficient branch-and-bound algorithms for MIQCPs.

Stability of Asymptotics of Christoffel–Darboux Kernels

We study the stability of convergence of the Christoffel–Darboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under ℓ 1 and random ℓ 2 diagonal perturbations. We also show that convergence to the sine kernel at x implies that μ({x}) = 0.

Stepping in Persons Poststroke: Comparison of Voluntary and Perturbation-Induced Responses

ObjectivesTo examine the stepping performance during voluntary and waist-pull perturbation-induced step initiation in people with chronic stroke. DesignRepeated-measures single-case design. SettingUniversity-based research laboratory. ParticipantsCommunity-dwelling stroke survivors (N=10). InterventionsNot applicable. Main Outcome MeasuresGround reaction forces and kinematic data were recorded to assess anticipatory postural adjustments (APAs) and step characteristics for both voluntary and induced stepping conditions. ResultsInduced stepping was performed with both the paretic (35% trials) and nonparetic legs (65% trials). Induced first steps occurred earlier and were executed faster than rapid voluntary steps. Compared with voluntary stepping, induced first step APAs were shorter in duration. Step height was higher with the nonparetic leg for both stepping conditions. Use of the paretic leg increased (52%) during the diagonal perturbations that passively unloaded the stepping limb compared with the use of the paretic leg (33%) for forward perturbations. ConclusionsThe results indicated differences in executing voluntary and induced stepping, and between the paretic and nonparetic limbs in individuals with chronic stroke. The findings suggested guidelines for using stepping as a component of neurorehabilitation programs for enhancing balance and mobility. Additional larger-scale studies remain to be undertaken to further investigate these issues.

One Step, Two Steps, Three Steps More … Directional Vulnerability to Falls in Community-Dwelling Older People

Falls leading to disability are common occurrences with advancing age. Stepping is a natural protective option for maintaining balance and preventing falls. There are directionally dependent challenges for protective stepping associated with falls among older individuals. The aim of this study was to determine the stepping response patterns evoked by different directions of externally applied postural disturbances in younger and older adults and in relation to falls. Seventy-five community-dwelling adults were tested: 26 younger adults and 49 older adults. Fall history of older participants was tracked prospectively for 1 year after testing. Steps were randomly evoked in 12 directions by a motorized waist-pull system. The number of recovery steps, type of stepping strategy, and first step kinematic characteristics were determined. Younger participants mainly used single recovery steps regardless of the perturbation direction. For the older groups, multiple steps occurred predominantly and were least for the forward-backward directions and greatest for the lateral directions. Trials with three or more recovery steps were increased laterally only for the fallers. Overall, fallers initiated stepping earliest, but other stepping characteristics were similar between the groups for forward-backward perturbations. Aging differences in stepping strategies for diagonal and lateral perturbations included numerous interlimb collisions. Adaptive changes in stepping characteristics between forward and lateral perturbations were also observed in relation to age and risk of falls. These results indicated an age-associated reduction in balance recovery effectiveness through stepping particularly for the lateral direction among older individuals at greater risk for falls.

The geometry of finite difference discretizations of semilinear elliptic operators

Discretizations by finite differences of some semilinear elliptic equations lead to maps , given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y − tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f′i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer–McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates.Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions.

Resonances for periodic Jacobi operators with finitely supported perturbations

We describe the resonances and the eigenvalues of a periodic Jacobi operator with finitely supported perturbations. In the case of small diagonal perturbations we determine their asymptotics.

A preconditioning technique for a class of PDE-constrained optimization problems

We investigate the use of a preconditioning technique for solving linear systems of saddle point type arising from the application of an inexact Gauss---Newton scheme to PDE-constrained optimization problems with a hyperbolic constraint. The preconditioner is of block triangular form and involves diagonal perturbations of the (approximate) Hessian to insure nonsingularity and an approximate Schur complement. We establish some properties of the preconditioned saddle point systems and we present the results of numerical experiments illustrating the performance of the preconditioner on a model problem motivated by image registration.