The link between age and migration propensity is long established, but existing models of country-level net migration ignore the effect of population age distribution on past and projected migration rates. We propose a method to estimate and forecast international net migration rates for the 200 most populous countries, taking account of changes in population age structure. We use age-standardized estimates of country-level net migration rates and in-migration rates over quinquennial periods from 1990 through 2020 to decompose past net migration rates into in-migration rates and out-migration rates. We then recalculate historic migration rates on a scale that removes the influence of the population age distribution. This is done by scaling past and projected migration rates in terms of a reference population and period. We show that this can be done very simply, using a quantity we call the migration age structure index (MASI). We use a Bayesian hierarchical model to generate joint probabilistic forecasts of total and age- and sex- specific net migration rates over five-year periods for all countries from 2020 through 2100. We find that accounting for population age structure in historic and forecast net migration rates leads to narrower prediction intervals by the end of the century for most countries. Also, applying a Rogers & Castro-like migration age schedule to migration outflows reduces uncertainty in population pyramid forecasts. Finally, accounting for population age structure leads to less out-migration among countries with rapidly aging populations that are forecast to contract most rapidly by the end of the century. This leads to less drastic population declines than are forecast without accounting for population age structure.

The optomechanical cavity (OMC) system has been a paradigm in the manifestation of continuous variable quantum information over the past decade. This paper investigates how quantum phase synchronization relates to bipartite Gaussian entanglement in coupled gain-loss mechanical oscillators, where the gain and loss rates are engineered by driving the cavity with blue and red detuned lasers, respectively. We examine the role of exceptional point in a deterministic way of producing self-sustained oscillations that induce robust quantum correlations among quadrature fluctuations of the oscillators. Particularly, steady phase synchronization dynamics along with the entanglement phenomena are observed in the effective weak coupling regime above a critical driving power. These phenomena are further verified by observing the mechanical squeezing and phase space rotations of the Wigner distributions. Additionally, we discuss how the oscillators frequency mismatches and decoherence due to thermal phonons impact the system dynamics. These findings hold promise for applications in phonon-based quantum communication and information processing.

This paper introduces a framework for solving alternating current optimal power flow (ACOPF) problems using graphics processing units (GPUs). While GPUs have demonstrated remarkable performance in various computing domains, their application in ACOPF has been limited due to challenges associated with porting sparse automatic differentiation (AD) and sparse linear solver routines to GPUs. We address these issues with two key strategies. First, we utilize a single-instruction, multiple-data abstraction of nonlinear programs. This approach enables the specification of model equations while preserving their parallelizable structure and, in turn, facilitates the parallel AD implementation. Second, we employ a condensed-space interior-point method (IPM) with an inequality relaxation. This technique involves condensing the Karush--Kuhn--Tucker (KKT) system into a positive definite system. This strategy offers the key advantage of being able to factorize the KKT matrix without numerical pivoting, which has hampered the parallelization of the IPM algorithm. By combining these strategies, we can perform the majority of operations on GPUs while keeping the data residing in the device memory only. Comprehensive numerical benchmark results showcase the advantage of our approach. Remarkably, our implementations -- MadNLP.jl and ExaModels.jl -- running on NVIDIA GPUs achieve an order of magnitude speedup compared with state-of-the-art tools running on contemporary CPUs.

In a star-network of qubits interacting via Heisenberg interaction of XYZ-type, we demonstrate a logarithmic growth of the localizable bipartite peripheral entanglement with increasing periphery-size and vanishing xy-anisotropy. This feature disappears when xy-anisotropy becomes non-zero, exhibiting an anisotropy effect, which can be negated by taking the system out of equilibrium by a qubit-local magnetic field. In the large-center and the competing-center limits of the model, the behaviour of LBPE is qualitatively different from that of the large-periphery limit. Also, the bipartite peripheral entanglement computed via a partial trace-based approach behaves qualitatively similarly to the LBPE in the large periphery limit, while in the other two limits, it behaves differently. We further consider the generalized description of localizable entanglement using unsharp measurements, and demonstrate that the logarithmic growth of LBPE is present for all noise strengths in the large-periphery limit, while in the competing-center limit, it does not.

In our adjacent work, we developed a spectral comparison principle for compound cocycles generated by delay equations. It allows to derive frequency inequalities for the uniform exponential stability of such cocycles by means of their comparison with stationary problems. Such inequalities are hard to verify purely analytically, and in this work we develop approximation schemes to verify some of the arising frequency inequalities. Besides some general theoretical results, in applications we stick to the case of scalar equations. By means of the Suarez-Schopf delayed oscillator and the Mackey-Glass equations, we demonstrate applications of the theory to reveal regions in the space of parameters where the absence of closed invariant contours can be guaranteed. Since the frequency inequalities are robust, so close systems also satisfy them, we expect the method to actually imply the global stability, as in known finite-dimensional results utilizing variants of the closing lemma, which is still awaiting developments in infinite dimensions.

This entry reviews Rudolf Carnap's philosophical views on the quantum mechanics of his time. It also offers some thoughts on how Carnap might have reacted to some recent developments in the foundations of quantum mechanics.

This paper considers non-Abelian homology groups of a group diagram introduced as homotopy groups of a simplicial change. We prove a theorem stating that the non-Abelian homology groups of a group diagram are isomorphic to the homotopy groups of the homotopy colimit of a classifying space diagram, with the dimension shifted by 1. Bousfield and Kan proved an isomorphism between the homotopy groups of an Abelian simplicial group and the homology groups of this simplicial group. We generalize this to non-Abelian simplicial groups. We also develop a method for finding a non-zero homotopy group of smallest dimension for the homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, we obtain a criterion for the isomorphism of the first non-Abelian and Abelian homology groups.

In a circuit-based quantum computer, the computing is performed via the discrete-time evolution driven by quantum gates. Accurate simulation of continuoustime evolution requires a large number of quantum gates and therefore suffers from more noise. In this paper, we find that shallow quantum circuits are sufficient to qualitatively observe some typical quantum phenomena in the continuous-time evolution limit, such as resonant tunneling and localization phenomena. We study the propagation of a spin excitation in Trotter circuits with a large step size. The circuits are formed of two types of two-qubit gates, i.e. XY gates and controlled- Rx gates, and single-qubit Rz gates. The configuration of the Rz gates determines the distribution of the spin excitation at the end of evolution. We demonstrate the resonant tunneling with up to four steps and the localization phenomenon with dozens of steps in Trotter circuits. Our results show that the circuit depth required for qualitative observation of some significant quantum phenomena is much smaller than that required for quantitative computation, suggesting that it is feasible to apply qualitative observations to near-term quantum computers. We also provide a way to use the physics laws to understand the error propagation in quantum circuits.

Post-selection inference has recently been proposed as a way of quantifying uncertainty about detected changepoints. The idea is to run a changepoint detection algorithm, and then re-use the same data to perform a test for a change near each of the detected changes. By defining the p-value for the test appropriately, so that it is conditional on the information used to choose the test, this approach will produce valid p-values. We show how to improve the power of these procedures by conditioning on less information. This gives rise to an ideal selective p-value that is intractable but can be approximated by Monte Carlo. We show that for any Monte Carlo sample size, this procedure produces valid p-values, and empirically that noticeable increase in power is possible with only very modest Monte Carlo sample sizes. Our procedure is easy to implement given existing post-selection inference methods, as we just need to generate perturbations of the data set and re-apply the post-selection method to each of these. On genomic data consisting of human GC content, our procedure increases the number of significant changepoints that are detected from e.g. 17 to 27, when compared to existing methods.

This article presents a constrained policy optimization approach for the optimal control of systems under nonstationary uncertainties. We introduce an assumption that we call Markov embeddability that allows us to cast the stochastic optimal control problem as a policy optimization problem over the augmented state space. Then, the infinite-dimensional policy optimization problem is approximated as a finite-dimensional nonlinear program by applying function approximation, deterministic sampling, and temporal truncation. The approximated problem is solved by using automatic differentiation and condensed-space interior-point methods. We formulate several conceptual and practical open questions regarding the asymptotic exactness of the approximation and the solution strategies for the approximated problem. As a proof of concept, we provide a numerical example demonstrating the performance of the control policy obtained by the proposed method.

We report numerical results on solving constrained linear-quadratic model predictive control (MPC) problems by exploiting graphics processing units (GPUs). The presented method reduces the MPC problem by eliminating the state variables and applies a condensed-space interior-point method to remove the inequality constraints in the KKT system. The final condensed matrix is positive definite and can be efficiently factorized in parallel on GPU/SIMD architectures. In addition, the size of the condensed matrix depends only on the number of controls in the problem, rendering the method particularly effective when the problem has many states but few inputs and moderate horizon length. Our numerical results for PDE-constrained problems show that the approach is an order of magnitude faster than a standard CPU implementation. We also provide an open-source Julia framework that facilitates modeling (DynamicNLPModels.jl) and solution (MadNLP.jl) of MPC problems on GPUs.

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multi-qubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized GHZ, the generalized W, Dicke, and the generalized Dicke states. For the generalized GHZ and W states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multi-qubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.

Graph theory provides a convenient framework for modeling and solving structured optimization problems. Under this framework, the modeler can arrange/assemble the components of an optimization model (variables, constraints, objective functions, and data) within nodes and edges of a graph, and this representation can be used to visualize, manipulate, and solve the problem. In this work, we present a ${\tt Julia}$ framework for modeling and solving graph-structured nonlinear optimization problems. Our framework integrates the modeling package ${\tt Plasmo.jl}$ (which facilitates the construction and manipulation of graph models) and the nonlinear optimization solver ${\tt MadNLP.jl}$ (which provides capabilities for exploiting graph structures to accelerate solution). We illustrate with a simple example how model construction and manipulation can be performed in an intuitive manner using ${\tt Plasmo.jl}$ and how the model structure can be exploited by ${\tt MadNLP.jl}$. We also demonstrate the scalability of the framework by targeting a large-scale, stochastic gas network problem that contains over 1.7 million variables.

Identifying and characterizing quantum phases of matter in the presence of long range correlations and/or spatial disorder is, generally, a challenging and relevant task. Here, we study a generalization of the Kiteav chain with variable-range pairing and different site-dependence of the chemical potential, addressing commensurable and incommensurable modulations as well as Anderson disorder. In particular, we analyze multipartite entanglement (ME) in the ground state of the dirty topological wires by studying the scaling of the quantum Fisher information (QFI) with the system's size. For nearest-neighbour pairing the Heisenberg scaling of the QFI is found in one-to-one correspondence with topological phases hosting Majorana modes. For finite-range pairing, we recognize long-range phases by the super-extensive scaling of the QFI and characterize complex lobe-structured phase diagrams. Overall, we observe that ME is robust against finite strengths of spatial inhomogeneity. This work contributes to establish ME as a central quantity to study intriguing aspects of topological systems.

We propose a novel feasible-path algorithm to solve the optimal power flow (OPF) problem for real-time use cases. The method augments the seminal work of Dommel and Tinney with second-order derivatives to work directly in the reduced space induced by the power flow equations. In the reduced space, the optimization problem includes only inequality constraints corresponding to the operational constraints. While the reduced formulation directly enforces the physical constraints, the operational constraints are softly enforced through Augmented Lagrangian penalty terms. In contrast to interior-point algorithms (state-of-the art for solving OPF), our algorithm maintains feasibility at each iteration, which makes it suitable for real-time application. By exploiting accelerator hardware (Graphic Processing Units) to compute the reduced Hessian, we show that the second-order method is numerically tractable and is effective to solve both static and real-time OPF problems.

We study a property of dynamic optimization (DO) problems (as those encountered in model predictive control and moving horizon estimation) that is known as exponential decay of sensitivity (EDS). This property indicates that the sensitivity of the solution at stage $i$ against a data perturbation at stage $j$ decays exponentially with $|i-j|$. {Building upon our previous results, we show that EDS holds under uniform boundedness of the Lagrangian Hessian, a uniform second order sufficiency condition (uSOSC), and a uniform linear independence constraint qualification (uLICQ). Furthermore, we prove that uSOSC and uLICQ can be obtained under uniform controllability and observability. Hence, we have that uniform controllability and observability imply EDS.} These results provide insights into how perturbations propagate along the horizon and enable the development of approximation and solution schemes. We illustrate the developments with numerical examples.

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that for a given pair of nodes, the sensitivity of the primal-dual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the decay rate remains uniformly bounded; this allows us to characterize the sensitivity behavior of NLPs defined over subgraphs of infinite graphs. The theoretical developments are illustrated with numerical examples.

Nonlinear optimization problems are found at the heart of real-time operations of critical infrastructures. These problems are computationally challenging because they embed complex physical models that exhibit space-time dynamics. We propose modeling these problems as graph-structured optimization problems, and illustrate how their structure can be exploited at the modeling level (for parallelizing function/derivative computations) and at the solver level (for parallelizing linear algebra operations). Specifically, we present a restricted additive Schwarz scheme that enables flexible decomposition of complex graph structures within an interior-point algorithm. The proposed approach is implemented as a general-purpose nonlinear programming solver that we call MadNLP.jl; this Julia-based solver is interfaced to the graph-based modeling package Plasmo.jl. The efficiency of this framework is demonstrated via problems arising in transient gas network optimization and multi-period AC optimal power flow. We show that our framework accelerates the solution (compared to off-the-shelf tools) by over 300%; specifically, solution times are reduced from 72.36 sec to 23.84 sec for the gas problem and from 515.81 sec to 149.45 sec for the power flow problem.

We present a general graph-based modeling abstraction for optimization that we call an OptiGraph. Under this abstraction, any optimization problem is treated as a hierarchical hypergraph in which nodes represent optimization subproblems and edges represent connectivity between such subproblems. The abstraction enables the modular construction of highly complex models in an intuitive manner, facilitates the use of graph analysis tools (to perform partitioning, aggregation, and visualization tasks), and facilitates communication of structures to decomposition algorithms. We provide an open-source implementation of the abstraction in the Julia-based package Plasmo.jl. We provide tutorial examples and large application case studies to illustrate the capabilities.

We study two-dimensional fermionic systems, displaying quadratic band touching in the normal state, in the presence of vortices and skyrmions of insulating and superconducting masses in the ordered phase. A prototypical example of such systems is the Bernal bilayer graphene that supports eight zero-energy modes in the presence of a mass vortex with the requisite U(1) symmetry. Near the vortex core, additional ten masses that close an SO(5) algebra can develop local expectation values by splitting the manifold of zero modes in five and ten different ways by lifting its SO(4) and SU(2) chiral symmetries, respectively. In particular, each SU(2) chiral symmetry can be broken by three distinct copies of chiral-triplet mass orders, giving rise to the notion of the color degeneracy among the competing orders near the vortex core. By contrast, a skyrmion of three anticommuting masses supports additional six masses in its core, and possesses an SU(2) isospin quantum number, besides the usual generalized U(1) charge. Consequently, charge $4e$ Kekulé pair density waves can develop in the skyrmion core of Néel layer antiferromagnet, while a skyrmion of quantum spin Hall insulator in addition supports an $s$ wave pairing. We also analyze the internal algebra of competing orders in the core of these defects on checkerboard or Kagome lattice that supports only a single copy of quadratic band touching in the normal state.

We analyze a time-coarsening strategy for model predictive control (MPC) that we call diffusing-horizon MPC. This strategy seeks to overcome the computational challenges associated with optimal control problems that span multiple timescales. The coarsening approach uses a time discretization grid that becomes exponentially more sparse as one moves forward in time. This design is motivated by a recently established property of optimal control problems that is known as exponential decay of sensitivity. This property states that the impact of a parametric perturbation at a future time decays exponentially as one moves backward in time. We establish conditions under which this property holds for a constrained MPC formulation with linear dynamics and costs. Moreover, we show that the proposed coarsening scheme can be cast as a parametric perturbation of the MPC problem and thus the exponential decay condition holds. We use a heating, ventilation, and air conditioning plant case study with real data to demonstrate the proposed approach. Specifically, we show that computational times can be reduced by two orders of magnitude while increasing the closed-loop cost by only 3%.

We compute the tree-level and one-loop matching relations for leading power gluon transverse momentum dependent parton distribution functions. At tree-level, working within the spinor formalism, we focus on twist-2 and twist-3 contributions, deriving the complete series of mass corrections for both T-even and T-odd distributions. At one-loop accuracy, we extend the parton-in-parton framework to include contributions beyond the leading term in the small-b expansion. Applying this methodology to the gluon sector, we obtain for the first time the Wandzura-Wilczek approximation for the gluon worm-gear T distribution. Furthermore, we develop a method to include the mass corrections in one-loop results and provide a closed-form expression for the mass series suitable for numerical implementations.

Organizational cybersecurity policies are often examined to determine whether they adequately comply standard security controls. This task is difficult because control statements are abstract, whereas policy documents describe governance practices in varied natural language. As a result, policy-based control assessment is time-consuming, difficult to standardize, and often difficult to document in a traceable manner. To address this gap, we present PROPARAG, an audit support approach for evaluating organizational cybersecurity policies against security controls autonomously. For each control, the approach retrieves relevant policy evidence, assesses coverage, identifies missing elements, and generates supporting explanations and recommendations. We evaluate PROPARAG on two real-world organizational policy corpora using 1,007 NIST SP 800-53 controls across both closed-source and open-source large language models (LLMs). The framework achieves F1 scores of 88.54 on OrgA and 82.31 on OrgB. The evaluation also shows that PROPARAG identifies relevant gaps in documented organizational policies and generates grounded recommendations for each identified gap. This research provides foundation for LLM-powered autonomous control-level assessment of organizational cybersecurity policies.

We investigate the long-term dynamical evolution of Ter2, Ter4, and Ter5, focusing on their mutual interactions, mass-loss behaviour, and survivability in the dense Galactic centre environment. We performed a suite of high-resolution direct N-body simulations over 8 Gyr, modelling three individual clusters that we also modelled as combined systems. We compared reference runs of isolated clusters with simulations of the full three-cluster system to quantify possible differences in mass loss, potential energy, and orbital behaviour. Our simulations reveal multiple close encounters between the Terzan clusters. The most significant encounters occur between Ter2-Ter4 and Ter4-Ter5, with their tidal radii exceeding the minimum separation. A notable case is the pair Ter2-Ter4, which approaches within 10 pc at a relative velocity of ~320 km/s. We found that the mass-loss rate is higher for the low-mass Ter2 and Ter4 systems in the combined three-cluster simulations than in our similar isolated runs, highlighting the importance of mutual cluster interactions. The common run clearly demonstrates that mutual gravitational interactions between clusters drive significant triaxial deformations, especially for Ter2 and Ter5, which evolve from nearly spherical to distinctly prolate shapes. In contrast, the isolated runs show clusters that remained almost perfectly spherical, confirming that the observed shape changes are correlated with the mutual interactions. The survivability and dynamical evolution of Galactic centre globular clusters cannot be fully understood without accounting for collective interactions among all systems within a few kiloparsecs. Our results emphasise the necessity of complex multi-cluster modelling in realistic Galactic potentials to capture the long-term fate of surviving and dissolved clusters

Detector characterization and instrumentation testing are often performed at cyclotron and synchrotron facilities, many of which were originally developed for medical applications in cancer therapy. For particle physics experiments requiring a single-particle resolution, pileup can significantly degrade data quality, making precise knowledge of the beam time structure essential for selecting appropriate readout parameters. However, such information is often unavailable from the facilities and challenging to determine experimentally. Here, we report measurements of the spill time structure at two medical accelerator facilities using a silicon carbide (SiC) particle sensor coupled to a high-frequency readout system. Owing to its high carrier saturation velocity and the tolerance to large bias voltages, SiC is well suited for fast readout and measurements requiring precise timing. Using a 6 GHz readout with custom SiC diodes, we characterize the micro-spill structure of both cyclotron and synchrotron beams on a sub-nanosecond timescale. The measured arrival-time distributions exhibit modulation with the accelerator RF frequencies, reflecting features of the extraction process. The resolved micro-spill structure enables quantitative estimation of pileup contributions and provides design constraints for future readout electronics. The presented results emphasize the importance of the characterization of the beam time-structure characterization for the development of precise readout systems.

In this article, the posinormality and coposinormality of weighted composition-differentiation operators on Hardy space $H^2(\mathbb{D})$ are investigated. It is observed that while a composition-differentiation operator $D_{ϕ,n}$ fails to be posinormal, the weighted composition-differentiation operator $D_{ψ,ϕ,n}$ can be posinormal for specific choices of $ψ, ϕ$. Some necessary conditions are obtained for posinormality and coposinormality of the operator $D_{ψ,ϕ,n}$. Furthermore, the adjoint formula for this operator is derived which also helped us to examine some results regarding posinormality of this operator.

The radii polynomial approach is an a posteriori validation method based on the contraction of a quasi-Newton operator. We apply this strategy to give a computer-assisted proof of a transverse heteroclinic orbit in the Shimizu--Morioka system, validating the equilibria and eigenpairs, the local invariant manifolds via the parameterization method, and the connecting orbit via a boundary-value problem. For each subproblem we present a four-step procedure: $(i)$ zero-finding formulation, $(ii)$ approximate zero, $(iii)$ approximate inverse, and $(iv)$ bound estimates. This highlights the unifying structure behind the a posteriori validation method. Alongside the analysis, we include code snippets implemented in Julia using the RadiiPolynomial library.

The emergence of a hantavirus variant aboard a commercial cruise ship presents a significant public health concern. This study develops a discrete-time stochastic Susceptible-Exposed-Infectious-Recovered-Dead model to estimate transmission dynamics, hidden exposed infections, and outbreak risk among passengers and crew. Epidemiological parameters and latent disease states were inferred using an Ensemble Adjustment Kalman Filter calibrated to reported case data from WHO and ECDC situation reports. The estimated basic reproduction number was 2.76, with a 95\% confidence interval of 2.52-2.99, indicating substantial potential for sustained onboard transmission before strict quarantine measures. Simulations further suggest that several exposed individuals may remain unidentified during the early outbreak phase, creating a hidden reservoir that symptom-based surveillance alone may fail to detect. These findings highlight the importance of rapid surveillance, widespread testing, targeted quarantine, and active monitoring of exposed individuals in confined travel settings. The proposed modeling framework can support timely outbreak assessment and intervention planning for infectious-disease events in similarly dense and spatially constrained populations.

In this paper the category of opposite brace triples is introduced in a general braided monoidal setting. Under cocommutativity, it is proved to be isomorphic to the category of Hopf braces. Furthermore, if one considers the subcategories arising from fixing one of the underlying Hopf algebras, then these two categories are also isomorphic to the category of matched pairs over that Hopf algebra.

Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C} \cap \mathcal{Q}$ is even and $\mathcal{C} \cap \mathcal{Q}$ contain at least one smooth point $z_o$ of $\mathcal{Q}$. In this paper, for every $\mathcal{Q}$ we find all possible conics $\mathcal{C}$ as above via studying geometry of $\mathcal{C}$ and $\mathcal{Q}$ through that of integral sections of a rational elliptic surface which canonically arises from $\mathcal{Q}$ and $z_o \in \mathcal{C} \cap \mathcal{Q}$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mathcal{Q}$, $\mathcal{C}$ and line passing through two of the singular points of $\mathcal{Q}$ .