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Bounded topologies are considered for functional differential equations of the neutral type in which present dynamics of the system are influenced by its past behavior. A special bounded topology is generated on a collection of absolutely continuous functions with essentially bounded derivatives, and an application to a class of nonlinear neutral functional differential equations due to Driver (1965) is presented.
Ordinary differential equations in linear topological space extended from Fattorini investigation, emphasizing use of linear operator
Over the past decade, it has been found that concepts from differential topology such as foliation and invariant distribution play a crucial role in the study of nonlinear systems. These concepts were first employed in studies of nonlinear controllability and observability. More recently, they have occurred in the study of decoupling and linearization via feedback. In connection with the widening area of application, requirements for a greater precision have arisen. The present paper represents an attempt to provide that precision. In this paper, an introduction is given of the basic concepts which are needed for an understanding of controllability, observability, and decoupling of nonlinear systems. Attention is given to mathematical preliminaries, aspects of invariance, nonlinear controllability and observability, disturbance decoupling, and controllability distributions.
The flux coordinates with dual-region safety factor (q) in the poloidal direction are developed in this work. The X-point effects on the ideal MHD modes in tokamaks are then analyzed using this coordinate system. Since the X-point effects mainly affect the edge region, the modes localized at the tokamak edge are particularly examined. Two types of modes are studied. The first is related to the conventional peeling or peeling-ballooning modes. The mode existence aligned with the local magnetic field in the poloidally core region as observed experimentally is confirmed. The X points are shown to contribute to a stabilizing effect for the conventionally treated modes with the surface-averaged q and with the tokamak edge portion truncated. The other is the axisymmetric modes localized in the vicinity of X points, which can affect the cross-field-line transport near the X points. The existence of axisymmetric modes points to the possibility of applying a toroidally axisymmetric resonant magnetic perturbation (RMP) in the X-point area for mitigating the edge localized modes, which can be an alternative to the current RMP design. The dual q description also has important implications for the existing non-axisymmetric RMP concept. It helps to understand why the RMP suppression of edge localized modes is difficult to achieve in the double-null tokamak configurations and points to the possibility of further improving the current RMP concept by considering the alignment to the local q.
In supersymmetric Yang–Mills theories tension-degenerate domain walls are typical. Adding matter fields in fundamental representation, we arrive at supersymmetric quantum chromodynamics (SQCD) supporting similar walls. We demonstrate that the degenerate domain walls can belong to one of two classes: i) locally distinguishable, i.e. those which differ from each other locally (which could be detected in local measurements); and ii) those which have identical local structure and are differentiated only topologically, through a judicially chosen compactification of R 4 . Depending on the number of flavors F and the pattern of Higgsing, both classes can coexist among SQCD k walls interpolating between the vacua n and n + k . We prove that the overall multiplicity of the domain walls obtained after accounting for both classes is ν N , k walls = N ! / [ ( N − k ) ! k ! ] , as was discovered previously in limiting cases. (Here, N is the number of colors.) Thus, ν N , k walls is a peculiar index. For the locally distinguishable degenerate domain walls, we observe two-wall junctions, a phenomenon specific for supersymmetry with central extensions. This phenomenon does not exist for topological replicas.
Principal component analysis (PCA) is a powerful method that can identify patterns in large, complex data sets by constructing low-dimensional order parameters from higher-dimensional feature vectors. There are increasing efforts to use space-and-time-dependent PCA to detect transitions in nonequilibrium systems that are difficult to characterize with equilibrium methods. Here, we demonstrate that feature vectors incorporating the position and velocity information of driven skyrmions moving through random disorder permit PCA to resolve different types of disordered skyrmion motion as a function of driving force and the ratio of the Magnus force to the dissipation. Since the Magnus force creates gyroscopic motion and a finite Hall angle, skyrmions can exhibit a greater range of flow phases than what is observed in overdamped driven systems with quenched disorder. We show that in addition to identifying previously known skyrmion flow phases, PCA detects several additional phases, including different types of channel flow, moving fluids, and partially ordered states. Guided by the PCA analysis, we further characterize the disordered flow phases to elucidate the different microscopic dynamics and show that the changes in the PCA-derived order parameters can be connected to features in bulk transport measures, including the transverse and longitudinal velocity-force curves, differential conductivity, topological defect density, and changes in the skyrmion Hall angle as a function of drive. We discuss how asymmetric feature vectors can be used to improve the resolution of the PCA analysis, and how this technique can be extended to find disordered phases in other nonequilibrium systems with time-dependent dynamics.
Based on new observations, a theoretical model of magnetic-field related heating processes in the solar corona is given. In this model, field-aligned currents are induced along coronal loops in thin current sheaths. Excitation of instabilities involving magnetic reconnection converts the energy associated with the current-related magnetic field directly into particle energy, where the heating process proceeds via short bursts corresponding to an intermittent disruption of the current sheath configuration. Because of the relatively low transverse thermal conduction, only a small fraction of the loop volume is heated to a much higher temperature than the average value. This is consistent with experimental observations of low filling factors of hot plasmas in coronal loops. Thus the model involves a repeated sequence of dynamic events taking into account the observed loop topology, the differential emission measure distribution in the 10 exp 6 - 10 exp 7 K range, the energy balance requirements in the loop, and the probable duty cycles involved in the heating processes.
We study the design of an electromagnetic cloak from multiple materials with an additional constraint on the mass of the cloak. Our problem is an example of a topology optimization problem, and we formulate this problem as a mixed-integer partial-differential equation constrained optimization (MIPDECO) problem, where Maxwell’s equation models the propagation of the wave through the cloak and surrounding medium. We use binary variables to model the assignment of the different materials, and their relevant properties (permittivity and density). The mass constraint adds a nontrivial constraint to this problem. We propose a two-phase strategy to solve this problem. In the first phase, we solve a continuous relaxation, and then propose a new variant of the feasibility pump that exploits the structure of the PDE to obtain an initial integral solution candidate. In the second phase, we use a trust-region approach to improve this incumbent. We also consider a continuation or mesh-sequencing approach to find better solutions faster on consecutively finer meshes. We present detailed numerical results to illustrate the effectiveness of our approaches for constructing multi-material cloaks with a mass constraint.
Approximation technique for flow graph that eliminates nonessential equivalent circuit elements, providing for derivation of most concise model with preassigned accuracy
This work presents single-differential electron-neutrino charged-current cross sections on argon measured using the MicroBooNE detector at the Fermi National Accelerator Laboratory. The analysis uses data recorded when the Neutrinos at the Main Injector beam was operating in both neutrino and antineutrino modes, with exposures of $2 \times 10^{20}$ and $5 \times 10^{20}$ protons on target, respectively. A selection algorithm targeting electron-neutrino charged-current interactions with at least one proton, one electron, and no pions in the final topology is used to measure differential cross sections as a function of outgoing electron energy, total visible energy, and opening angle between the electron and the most energetic proton. The interaction rate as a function of proton multiplicity is also reported. The total cross section is measured as [4.1 $\pm$ 0.4 (stat.) $\pm$ 1.2 (syst.)]$ $$\times 10^{-39} \mathrm{cm}^{2}/ \mathrm{nucleon}$. The unfolded cross-section measurements are compared to predictions from neutrino event generators commonly employed in the field. Good agreement is seen across all variables within uncertainties.
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Nonlinear Volterra integral equation, extending topological dynamics application of nonautonomous ordinary differential equations
Topologies introduced on state space for differential equations to obtain dynamical systems
The paper develops a model for the anomalous viscosity in accretion disks based on the hypothesis that the hydrodynamic turbulence within the disk takes the form of spatially localized magnetic flux cells. The local shear flow due to Keplerian differential rotation distorts the flux cell topology, converting shear flow energy into magnetic energy. In the radial diffusion approximation, the kinematic viscosity is estimated from the radial displacement and is shown to maximize at flux cell scale lengths for which the shear flow stopping and reconnection times are equal.
Here, the power flow equations are important in numerous power systems problems of practical interest which consider alternating current power flow (ACPF) physics. Perhaps the most well studied being the alternating current optimal power flow problem (ACOPF), seeking to optimize the operation of an electric power system. Due to their non-linearity, problems which include the power flow equations are typically challenging, particularly in optimization. Interestingly, the set of solutions to the power flow equations forms a smooth manifold. As a result, differential geometry can be used to describe and analyze this set of equations. This approach has proven effective in several engineering applications (e.g., solving ACOPF and analyzing the solution space boundary). Central to the success of this approach is an understanding of the power flow manifold's geometry. In this work, we develop the geometric and topological properties of this manifold using concepts from differential geometry. After demonstrating the convenience of this manifold's representation as a function's graph, computational methods are emphasized: we develop retractions, error bounds for linear approximation, and formulas for evaluating the Riemannian metric (including associated objects such as geodesics and the curvature tensor). Scalar curvature and the second fundamental form play a new role in quantifying the quality of linear approximations, like the popular direct current approximation. All functions are implemented in Julia and available in an online repository. Proofs are included for completeness.
As a part of the Proton Improvement Plan – II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
As a part of the Proton Improvement Plan II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
The state of the art in topology and flow geometry is presented. Solution techniques for partial differential equations are reviewed and included developments in coordinate transformations, conformal mapping, and invariant imbeddings. Applications of these techniques in fluid mechanics, flow geometry, boundary value problems, and fluidics are presented.