Explicit Conditions Research Articles

Adjoints and canonical forms of tree amplituhedra

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm {Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

Explicit Commutativity for Lamé Linear Time‐Varying Differential Systems

ABSTRACTThis article studies the commutativity and sensitivity of the Lamé linear time‐varying systems (LTVSs), and investigates the effects of disturbances on such systems. The commutative pair for the Lamé LTVS of order is found, that is, a new Lamé LTVS of order is derived using the explicit commutative theories for zero initial conditions (ICs). For the case of nonzero ICs, the commutativity between the connected input–output of Lamé systems and is studied. New and simple explicit commutative theories and conditions for second‐order LTVSs are derived, simplifying the use of commutativity for practical and industrial scenarios. These findings enable us to analyse the commutativity, sensitivity, robustness and stability of Lamé systems, and to determine the effects of disturbances. The explicit results presented in this article are supported by simulations and verified by examples and constitute a significant contribution to science and engineering applications.

Shaped pattern synthesis for hybrid analog–digital arrays via manifold optimization-enabled block coordinate descent

Hybrid analog–digital (HAD) architecture is a promising means to realize large-scale arrays owing to the judicious trade-off between system performance and hardware complexity. This paper investigates the beampattern shaping of HAD arrays through minimizing the beampattern matching error between desired and actual patterns. Due to the nonconvex constraints on analog and digital weights and the nonconvex nonsmooth objective function, the resultant codesign is NP-hard. To address this issue, we create an efficient algorithm by leveraging block coordinate descent (BCD) and Riemannian manifold optimization. We first equivalently represent the objective function as a smooth bi-quadratic function by introducing a uni-modulus auxiliary variable. Subsequently, we alternatingly optimize the scaling factor, digital weights, analog weights and auxiliary variable under the BCD framework, where the simultaneous update of analog weights and auxiliary variable is recast as uni-modular constrained quadratic programming which is efficiently solved by Riemannian Newton method, and the digital weights have a global optimal solution via Lagrangian multiplier method. We also derive an explicit convergence condition of this algorithm. Numerical results demonstrate that the proposed algorithm has superior performance and faster convergence speed than alternative algorithms, and produces nearly the same mainlobe gain as fully digital arrays with significantly fewer radio-frequency chains.

Effective extensional–torsional elasticity and dynamics of helical filaments under distributed loads

We study slender, helical elastic rods subject to distributed forces and moments. Focussing on the case when the helix axis remains straight, we employ the method of multiple scales to systematically derive an ‘equivalent-rod’ theory from the Kirchhoff rod equations: the helical filament is described as a naturally-straight rod (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Importantly, our analysis is asymptotically exact in the limit of a ‘highly-coiled’ filament (i.e., when the helical wavelength is much smaller than the characteristic lengthscale over which the filament bends due to external loading) and is able to account for large, unsteady displacements. In addition, our analysis yields explicit conditions on the external loading that must be satisfied for a straight helix axis. In the small-deformation limit, we exactly recover the coupled wave equations used to describe the free vibrations of helical coil springs, thereby justifying previous equivalent-rod approximations in which linearised stiffness coefficients are assumed to apply locally and dynamically. We then illustrate our theory with two loading scenarios: (I) a heavy helical rod deforming under its own weight; and (II) the dynamics of axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. In both scenarios, we demonstrate excellent agreement with solutions of the full Kirchhoff rod equations, even beyond the formal limit of validity of the ‘highly-coiled’ assumption. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, and yields analytical insight into their elastic instabilities. In particular, our analysis indicates that tensile instabilities are a generic phenomenon when helical rods are subject to a combination of distributed forces and moments.

Isolated Calmness of Perturbation Mappings and Superlinear Convergence of Newton-Type Methods.

In this paper, we characterize Lipschitzian properties of different multiplier-free and multiplier-dependent perturbation mappings associated with the stationarity system of a so-called generalized nonlinear program popularized by Rockafellar. Special emphasis is put on the investigation of the isolated calmness property at and around a point. The latter is decisive for the locally fast convergence of the so-called semismooth* Newton-type method by Gfrerer and Outrata. Our central result is the characterization of the isolated calmness at a point of a multiplier-free perturbation mapping via a combination of an explicit condition and a rather mild assumption, automatically satisfied e.g. for standard nonlinear programs. Isolated calmness around a point is characterized analogously by a combination of two stronger conditions. These findings are then related to so-called criticality of Lagrange multipliers, as introduced by Izmailov and extended to generalized nonlinear programming by Mordukhovich and Sarabi. We derive a new sufficient condition (a characterization for some problem classes) of nonexistence of critical multipliers, which has been also used in the literature as an assumption to guarantee local fast convergence of Newton-, SQP-, or multiplier-penalty-type methods. The obtained insights about critical multipliers seem to complement the vast literature on the topic.

Adapting spatiotemporal gait symmetry to functional electrical stimulation during treadmill walking.

Individuals with neurological impairments often exhibit asymmetrical gait patterns. This study explored the potential of using functional electrical stimulation (FES) as a perturbation method during treadmill walking to promote gait symmetry adaptation by investigating whether the FES perturbation could induce gait adaptation concerning spatial and temporal gait symmetry in healthy subjects. In the FES perturbation, both legs received electrical pulses at the same period as the subjects' initial stride duration, and the temporal gap between the two pulses for each leg was manipulated over a 7-min period. Following this, subjects continued to walk for another 5 minutes without FES. Subjects participated in two trials: implicit and explicit. In the implicit trial, they walked comfortably during FES perturbation without consciously adjusting their gait. In the explicit trial, they voluntarily synchronized their toe-off phase to the stimulation timing. To examine the effects of the FES perturbation, we measured step length and stance time and then analyzed changes in step length and stance time symmetries alongside their subsequent aftereffects. During the explicit trial, subjects adapted their gait patterns to the electrical pulses, resulting in a directional change in stance time (temporal) symmetry, with the left stance becoming shorter than the right. The stance time asymmetry induced by FES perturbation showed a slight residual effect. In the implicit trial, the directional change trend was slightly observed but not statistically significant. No consistent trend in step length (spatial) symmetry changes was observed in either condition, indicating that subjects may adapt their spatial gait patterns independently of their temporal patterns. Our findings suggest that the applied FES perturbation strategy under explicit condition can induce adaptations in subjects' temporal gait asymmetry, particularly in stance. The implicit condition showed a similar slight trend but was not statistically significant. Further experiments would provide deeper understanding into the mechanism behind subjects' response to FES perturbations, as well as the long-term effects of these perturbations on the spatial and temporal aspects of gait symmetry.

Use your imagination for better performance. Effects of analogy instruction in motor skills. A systematic review and meta-analysis

This systematic review and meta-analysis examined the effects of analogy instruction (ANA) on motor performance and knowledge declared (KD) compared with explicit learning (EXP) and control conditions. Five databases were included. The study analyzed 16 randomized controlled trials. Subsequent analysis was performed for moderators variables as age, skill, retention, stress situations number of rules, specificity and number of trials. The ANA instruction demonstrated greater efficacy than the control (ES = 0.32, p = 0.03) or EXP condition (ES = 0.29, p = 0.02) in motor tasks performance in general terms. ANA instructions also showed superiority in motor performance when compared to control conditions in retention (ES = 5.72, p = 0.004), and a trend towards significance was found under stress (ES = 1.18, p = 0.05). ANA also showed superiority in motor performance when compared to EXP instruction (ES = 0.29, p = 0.02). ANA demonstrated greater effects than EXP in retention (ES = 7.25, p = 0.01), but not under stress (ES = 0.62, p = 0.18). Sub-analyses demonstrated that children (all p < 0.01) and novices (all p < 0.01) are more likely to benefit from ANA instruction when compared to control or EXP. A subgroup analysis based on quantity of information comparing ANA versus EXP condition shows that ANA is more effective for enhancing motor performance than EXP when the number of rules are similar. Sub-analyses comparing ANA versus CNT shows that as the number of repetitions increases and the task becomes less specific, ANA instruction significantly improves performance. A comparison between ANA and EXP indicates no significant differences in performance regarding the number of repetitions and task specificity. A secondary analysis examined KD of different instructions. KD was greater in EXP instructions (ES = −1.48, p < 0.001) when compared to ANA. Findings suggest that analogy instruction may improve motor performance, especially in novice and child populations. However, caution is needed due to concerns when comparing with other instructional types and environments, as well as due to high heterogeneity in most of the comparisons and high risk of bias in included studies.

Triple-diffusive instabilities in Ellis fluid-saturated porous layers: Dynamics of oscillatory convection

Triple-diffusive convection in Ellis fluid-saturated porous layers has a wide array of real-world applications, including enhanced oil recovery, optimized geothermal energy extraction, and improved food processing and drug delivery systems. It also plays a crucial role in environmental management, particularly in controlling groundwater contamination and maintaining soil health by modeling pollutant transport and nutrient dynamics. This study explores the onset of convection in an Ellis fluid-saturated porous layer, influenced by three stratifying agents with differing diffusivities. A modified Darcy porous medium, salted from below, is subjected to horizontal throughflow driven by a prescribed pressure gradient. Through normal mode analysis, a linear stability analysis is conducted, resulting in explicit threshold conditions for the onset of convection. The findings reveal that convection begins with oscillatory motion, driven by the combined effects of the pressure gradient and solute concentration gradients. Notably, the study uncovers the emergence of disconnected, closed, heart-shaped oscillatory neutral curves, indicating the presence of three critical values of the solutal Darcy-Rayleigh number required to establish linear instability criteria and novel discovery for an Ellis fluid-saturated porous medium. Moreover, the results show that increasing the solutal Darcy-Rayleigh number and the Ellis power-law index stabilizes the system, while a higher Darcy-Ellis number leads to destabilization. The results obtained in the limiting cases are found to be consistent with those reported in previous studies.

On nonemptiness of Newton strata in the $B_{\mathrm{dR}}^{+}$-Grassmannian for $\mathrm{GL}_{n}$

We study the Newton stratification in the B_{\mathrm{dR}}^{+} -Grassmannian for \mathrm{GL}_{n} associated to an arbitrary (possibly nonbasic) element of B(\mathrm{GL}_{n}) . Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in B(\mathrm{GL}_{n}) , our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues–Fontaine curve.

How Cells Stay Together: A Mechanism for Maintenance of a Robust Cluster Explored by Local and Non-local Continuum Models.

Formation of organs and specialized tissues in embryonic development requires migration of cells to specific targets. In some instances, such cells migrate as a robust cluster. We here explore a recent local approximation of non-local continuum models by Falcó et al. (SIAM J Appl Math 84:17-42, 2023). We apply their theoretical results by specifying biologically-based cell-cell interactions, showing how such cell communication results in an effective attraction-repulsion Morse potential. We then explore the clustering instability, the existence and size of the cluster, and its stability. For attractant-repellent chemotaxis, we derive an explicit condition on cell and chemical properties that guarantee the existence of robust clusters. We also extend their work by investigating the accuracy of the local approximation relative to the full non-local model.

On the nature of Bregman functions

Let C⊂Rn be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.

K-stability of Fano 3-folds of Picard rank 3 and degree 22

AbstractWe prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 22 that satisfy very explicit generality condition.

Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

تم في هذه الدراسة النظر في النظام التفاضلي التربيعي ثلاثي الأبعاد، حيث يصبح نقطة الأصل الإحداثيات نقطة توازن هوبف. تم دراسة وجود واستقرار الدورات النهائية التي تنبثق من نقطة هوبف. يتم حساب معاملات ليبانوف المرتبطة بنقطة هوبف باستخدام طريقة الإسقاط. أولاً، تم تحديد أربع عائلات من شروط المعلمات التي من خلالها يمكن للنظام التفاضلي التربيعي ثلاثي الأبعاد أن يظهر البعد الثالث لتشعب هوبف. تم إعطاء الدليل التحليلي لكل مجموعة من الحالات المعلمية عن طريق حساب معاملات ليبانوف، تصفير لقيمة معاملات ليبانوف الأولى و الثاني و غيرالصفري لمعاملات ليبانوف الثالثة. تم تقديم الشروط الواضحة لوجودية واستقرارية ثلاث دورات النهائية ناشئة عن كل عائلة من تشعبات هوبف. يُظهر ناتج الوجود نقطة هوبف مستقرة (غير مستقرة)، مصحوبة بظهور دورتين حديتين مستقرتين (غير مستقرة) جنبًا إلى جنب مع دورة حدية واحدة غير مستقرة (مستقرة) في جوار النقطة الأصل غير المستقر (المستقر) لنظام المعادلة التربيعية ثلاثية الأبعاد. بالإضافة إلى ذلك، يتم استخدام النتيجة لاستكشاف الدورات النهائية لنظام الجذب الفوضوي n-scroll، والذي له العديد من الاستخدامات العملية، بما في ذلك الاتصال الآمن، والتشفير، وتوليد الأرقام العشوائية، والروبوتات المتنقلة المستقلة. تم اشتقاق الشروط التي بموجبها تصبح نقطة الأصل لهذا النظام هي نقطة هوبف، ويمكن أن توجد ثلاث دورات نهائية حول نقطة هوبف. وأخيرًا، توضح العروض العددية أن النظام يخضع لتشعب هوبف فوق الحرج، مما يؤدي إلى دورتين حديتين مستقرتين وواحدة غير مستقرة. وعلاوة على ذلك، تم التحقق من كافة النتائج.

A complete analysis of a slow–fast modified Leslie–Gower predator–prey system

In this paper, we transform a modified Leslie–Gower predator–prey system into the corresponding fast–slow version by assuming that prey reproduces much faster than predator, and then perform a complete analysis about its dynamics. More specifically, we find the necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its (or their) location, and then we further fully determine its (or their) dynamics under explicit conditions. Besides, by converting the slow–fast system into its slow–fast normal form, we are able to characterize its rich dynamics completely, including relaxation oscillation, singular Hopf bifurcation, canard explosion, homoclinic orbits, heteroclinic orbits and global attraction of equilibrium. Moreover, the sufficient conditions to guarantee these various rich dynamics are explicitly given, including the explicit conditions to determine whether singular Hopf bifurcation is supercritical or subcritical, which generally cannot be explicitly derived in the existing literatures. Additionally, the cyclicity of diverse canard cycles is found under explicit conditions. Of particular interest is that we show the existence and uniqueness of one canard cycle without head whose cyclicity is at most two under explicit parameters conditions. Our results complement and enrich the previous work (relaxation oscillation, and global attraction of a boundary equilibrium) about this fast–slow system. We also employ numerical simulations to illustrate our theoretical analysis.

Children’s Metacognition and Their Second Language Acquisition: A Longitudinal Study

Abstract The present study aimed to examine the relationship between children’s metacognition and their later second language acquisition in implicit and explicit learning conditions, as well as the effects of time and condition on children’s metacognition and second language acquisition. Seventy Chinese children (M = 8.13 ; sd = 3.34) were followed from Year 2 to Year 3 at primary school. Children in Group 1 were in the Explicit Learning Condition, while children in Group 2 were in the Implicit and Explicit Learning Condition. A range of tasks were used to assess children’s metacognition and second language acquisition. Children’s implicit metacognitive skills and explicit metacognitive knowledge had significant predictive roles in their later second language acquisition. Children in Group 2 had higher levels of improvement in later implicit and explicit metacognition as well as in implicit and explicit second language acquisition than their counterparts in Group 1.

K -inflation: The legitimacy of the classical treatment

In this paper we consider a general theory of k-inflation and find out that it may be in a strong coupling regime. We derive accurate conditions of classical description validity using unitarity bounds for this model. Next, we choose a simple toy model of k-inflation and obtain the explicit condition, which guarantees that the generation of perturbations is performed in a controllable way, i.e., the exit from the effective horizon occurs in the weak coupling regime. However, for the same toy model the corresponding experimental bounds on a nonlinear parameter fNLequil associated with non-Gaussianities of the curvature perturbation provide much stronger constraint than strong coupling absence condition. Nevertheless, for other known models of inflation this may not be the case. Generally, one should always check if classical description is legitimate for chosen models of inflation. Published by the American Physical Society 2024

Balancing Equitable Enforcement and Property Rights: Revisiting Section 23 of the Maintenance and Welfare of Parents and Senior Citizens Act, 2007 of India

Abstract The Maintenance and Welfare of Parents and Senior Citizens Act, 2007 (hereinafter “MWPSC Act”) of India, specifically section 23, addresses the responsibility of heirs towards their parents’ maintenance and the consequences of failure in such obligations. This article critically evaluates the interpretation and application of section 23, focusing on the judicial reasoning, implications on property rights, and comparisons with international jurisprudence. The article proposes a balanced approach to ensure equitable enforcement while safeguarding third-party interests and the broader legal landscape. It advocates for a nuanced understanding of the statute, emphasizing the need for explicit conditions, limitations on resorting to section 23, and harmonizing maintenance obligations with property rights. Drawing from foreign legal systems, this article underscores the uniqueness of Indian legislation while highlighting the absence of similar provisions in other jurisdictions. By analysing the interpretation and application of section 23 of the MWPSC Act, this article contributes to the broader discourse on balancing legislative intent, public policy, familial obligations, and property rights in the context of elder care in India.

Lifting Directional Fields to Minimal Sections

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.

Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution. Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In this work, we first prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ℝ d for d≥3. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Second, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ℝ d . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.