Cone Theory Research Articles

On dual cone theory for Euclidean Bosonic equations

On dual cone theory for Euclidean Bosonic equations

Aerodynamic analysis of an airframe-inlet integrated full-waverider vehicle designed by osculating cone theory

Recently proposed, the airframe/inlet integrated full-waverider vehicle design methodology introduces an innovative approach to the design of air-breathing hypersonic vehicles. This methodology, compared to traditional integrated waverider design methods, leverages the features of the waverider to enhance the compression ability of the precompression surface and the lift-to-drag ratio characteristics of the vehicle. However, existing full-waverider design methods, based on cone-derived waverider theory, result in a gas cross-flow phenomenon at the inlet lip due to the conical shockwave. This paper suggests an airframe/inlet integrated full-waverider vehicle design method that utilizes the osculating cone waverider theory to improve gas intake quality. Initially, we present the method to obtain profile lines and basic flow field models at different osculating planes of the vehicle. We then validate the osculating cone integrated full-waverider vehicle design methodology through computational fluid dynamics. Although the viscous effect slightly reduces the lift-to-drag ratio of the vehicle and the total pressure recovery coefficient at the isolator exit, the shock wave remains attached to the leading-edge. Finally, a comparison with the cone-derived integrated full-waverider vehicle reveals that the osculating cone integrated full-waverider vehicle achieves similar aerodynamic performance, but with superior gas intake quality. The results indicate that the methodology proposed herein holds promise for guiding future engineering applications of integrated full-waverider vehicles.

Loss cone shielding

ABSTRACT A star wandering close enough to a massive black hole can be ripped apart by the tidal forces of the black hole. The advent of wide-field surveys at many wavelengths has quickly increased the number of tidal disruption events (TDEs) observed, and has revealed that (i) observed TDE rates are lower than theoretical predictions and (ii) E+A galaxies are significantly overrepresented. This overrepresentation further worsens the tension between observed and theoretically predicted TDEs for non-E+A galaxies. Classical loss cone theory focuses on the cumulative effect of many weak scatterings. However, a strong scattering can remove a star from the distribution before it can get tidally disrupted. Most stars undergoing TDEs come from within the radius of influence, the densest environments of the Universe. In such environments, close encounters rare elsewhere become non-negligible. We revise the standard loss cone theory to take into account classical two-body interactions as well as strong scattering, collisions, tidal captures, and study under which conditions close encounters can shield the loss cone. We (i) analytically derive the impact of strong scattering and other close encounters, (ii) compute time-dependent loss cone dynamics including both weak and strong encounters, and (iii) derive analytical solutions to the Fokker–Planck equation with strong scattering. We find that (i) TDE rates can be reduced to up to an order of magnitude and (ii) strong shielding preferentially reduces deeply plunging stars. We also show that stellar overdensities, one possible explanation for the E + A preference, can fail to increase TDE rates when taking into account strong scattering.

On the pollution of white dwarfs by exo-Oort cloud comets

ABSTRACT A large fraction of white dwarfs (WDs) have metal-polluted atmospheres, which are produced by accreting material from remnant planetary systems. The composition of the accreted debris broadly resembles that of rocky Solar system objects. Volatile-enriched debris with compositions similar to long-period comets (LPCs) is rarely observed. We attempt to reconcile this dearth of volatiles with the premise that exo-Oort clouds (XOCs) occur around a large fraction of planet-hosting stars. We estimate the comet accretion rate from an XOC analytically, adapting the ‘loss cone’ theory of LPC delivery in the Solar system. We investigate the dynamical evolution of an XOC during late stellar evolution. Using numerical simulations, we show that 1–30 per cent of XOC objects remain bound after anisotropic stellar mass-loss imparting a WD natal kick of ${\sim}1 \, {\rm km \, s^{-1}}$. We also characterize the surviving comets’ distribution function. Surviving planets orbiting a WD can prevent the accretion of XOC comets by the star. A planet’s ‘dynamical barrier’ is effective at preventing comet accretion if the energy kick imparted by the planet exceeds the comet’s orbital binding energy. By modifying the loss cone theory, we calculate the amount by which a planet reduces the WD’s accretion rate. We suggest that the scarcity of volatile-enriched debris in polluted WDs is caused by an unseen population of 10–$100 \, \mathrm{au}$ scale giant planets acting as barriers to incoming LPCs. Finally, we constrain the amount of volatiles delivered to a planet in the habitable zone of an old, cool WD.

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone. Cones that are homogeneous and self-dual are called symmetric. Conic optimization problems over symmetric cones have been extensively studied, particularly in the literature on interior-point algorithms, and as the foundation of modelling tools for convex optimization. In this paper we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual.We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. Chordal sparsity guarantees that positive define matrices in the cone have zero-fill Cholesky factorizations. The stronger properties that make the cone homogeneous guarantee that the inverse Cholesky factors have the same zero-fill pattern. We describe transitive subsets of the cone automorphism groups, and important properties of the composition of log-det barriers with the automorphisms.Next, we consider extensions to linear slices of the positive semidefinite cone, and review conditions that make such cones homogeneous. An important example is the matrix norm cone, the epigraph of a quadratic-over-linear matrix function. The properties of homogeneous sparse matrix cones are shown to extend to this more general class of homogeneous matrix cones.We then give an overview of the algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation.We conclude by discussing the role of homogeneous structure in primal–dual symmetric interior-point methods, contrasting this with the well-developed algorithms for symmetric cones that exploit the strong properties of self-scaled barriers, and with symmetric primal–dual methods for general convex cones.

Application of normal cones to the computation of solutions of the nonlinear Kolmogorov backward equation

A numerical approach to computing solutions of a generalized Kolmogorov backward equation is proposed, in which the stochastic matrix is replaced by a nonlinear operator obtained as the lower bound of a set of stochastic matrices. The equation is central to the theory of imprecise Markov chains in continuous time, which has made rapid progress in recent years. One of the obstacles to its implementation remains the high computational complexity, with the prevailing existing approaches relying on a discretization of the time interval. In order to achieve sufficient accuracy of the approximations, the grid must typically contain a large number of points on which optimization steps are performed, usually using linear programming. The main goal of this work is to develop a new, more efficient approach by significantly reducing the number of optimization steps required to achieve the prescribed accuracy of the solutions. Our approach is based on the Lipschitz continuity of the solutions of the equation with respect to time, which results in the optimization problems occurring at nearby points of the time interval having similar optimal solutions. This property is exploited using the theory of normal cones of convex polytopes. If the solution vectors remain within the same normal cone of a polytope corresponding to the nonlinear operator in a given interval, the optimization problem to be solved becomes linear, which allows much faster computations. This paper is primarily concerned with providing the theoretical basis for the new technique. However, initial tests show that it significantly outperforms existing methods in most cases.

A Sliding Mode Control Algorithm with Elementary Compensation for Input Matrix Uncertainty in Affine Systems

This paper aims to develop a sliding mode control (SMC) approach with elementary compensation for input matrix uncertainty in affine systems. As a multiplicative uncertainty regarding the control inputs, input matrix uncertainty adversely modifies the control effort and even further causes the instability of systems. To solve this issue, a sliding mode control algorithm is developed based on a two-step design strategy. The first step is to design a general sliding mode controller for the system without input matrix uncertainty. In the second step, a control term is specially designed to compensate for input matrix uncertainty. In order to realize the elementary compensation for input matrix uncertainty, this term is obtained by solving a nonlinear vector equation which is derived from the Lyapunov function inequality. Theorems and lemmas based on the convex cone theory are proposed to guarantee the existence and uniqueness of the solution to the vector equation. Additionally, an algorithmic process is proposed to solve the vector equation efficiently. In the simulation part, the proposed controller is applied to two systems with different structures and compared with two state-of-the-art SMC algorithms. The comprehensive simulation results demonstrate that the proposed method is able to provide the closed-loop system with a competitive performance in terms of convergence level, overshoot reduction and chattering suppression.

Thermal Elastohydrodynamic Lubrication Analysis for Rolling Cone Enveloping End Face Worm Drive

To explore the thermal elastohydrodynamic lubrication performance of rolling cone enveloping end face worm drive, the mathematical model of thermal elastohydrodynamic lubrication is established based on the meshing theory of rolling cone enveloping end face worm drive and elastohydrodynamic lubrication theory. With the help of the numerical analysis software MATLAB, the thermal elastohydrodynamic lubrication performance under smooth conditions is selectively analyzed by the multigrid method. Besides, the effects of throat coefficient, cone apex radius and half cone angle of the rolling cone on thermal elastohydrodynamic lubrication performance are analyzed at the meshing out of the transmission pair. The results show that the thermal elastohydrodynamic lubrication film thickness at meshing out of the worm gear is the smallest; the oil film thickness, oil film pressure and oil film temperature are reduced from meshing in to meshing out. The oil film temperature at the interlayer is the largest. The oil film temperature at the end face worm surface is the smallest. The small half-cone angle of the cone apex radius and larger throat coefficient and cone apex radius may achieve better thermal elastohydrodynamic lubrication performance of rolling cone enveloping end face worm drive.

A Numerical Investigation of Tidal Disruption Rate Enhancement in Post-Starburst galaxies

Tidal disruption is an uncommon but important phenomenon that occurs when a star’s gravity is overcome by that of the SMBH which it is orbiting. In this process, researchers may gleam various properties of the host galaxy’s nucleic region, such as the accretion process, debris fallback, behavior of active galactic nuclei and quasars, etc. In this work we present a numerical analysis of TDE rates using the loss cone theory, a leading theoretical picture in this field, based on changes to several pivotal variables, such as SMBH mass, stellar mass, velocity dispersion etc. We then extend this to fit the profiles of E+A post starburst galaxies and the occasion of binary black hole systems.

Conic Input Mapping Design of Constrained Optimal Iterative Learning Controller for Uncertain Systems.

In this article, we study the optimal iterative learning control (ILC) for constrained systems with bounded uncertainties via a novel conic input mapping (CIM) design methodology. Due to the limited understanding of the process of interest, modeling uncertainties are generally inevitable, significantly reducing the convergence rate of the control systems. However, huge amounts of measured process data interacting with model uncertainties can easily be collected. Incorporating these data into the optimal controller design could unlock new opportunities to reduce the error of the current trail optimization. Based on several existing optimal ILC methods, we incorporate the online process data into the optimal and robust optimal ILC design, respectively. Our methodology, called CIM, utilizes the process data for the first time by applying the convex cone theory and maps the data into the design of control inputs. CIM-based optimal ILC and robust optimal ILC methods are developed for uncertain systems to achieve better control performance and a faster convergence rate. Next, rigorous theoretical analyses for the two methods have been presented, respectively. Finally, two illustrative numerical examples are provided to validate our methods with improved performance.

$\alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations

In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha> 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.

Equivalence of cubical and simplicial approaches to (∞,n)-categories

We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for (n-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for (n-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.

Opportunity Cone Theory: The Physics of the Economic Management of Amalgamations of Spacetime and Matter

Opportunity Cone Theory: The Physics of the Economic Management of Amalgamations of Spacetime and Matter

Revealing the role of forests in the mobility of geophysical flows

Forests cover 39% of low-altitude mountainous areas globally, and they are considered natural barriers against geophysical flows. Owing to the complexity of flows, the protective effects of forests can hardly be considered in hazard management. In this study, the interactions between granular geophysical flows and forests are explored using a MPM-DEM simulator from a viewpoint of supersonic bow shocks. We show that bow shocks can be approximated by a non-linear analytical framework based on the Mach cone theory. To generate evidence of flow–forest interactions, a new experimental model forest is built. Then, a computational model of granular impact on trees is calibrated and solved by the MPM-DEM method. The results reveal that interactions between overlapping bow shocks considerably affect the flow mobility. Bow shocks either reduce flow momentum or concentrate momentum to increase the runout distance by a factor of up to 1.5 compared with the runout distance of a non-forested area. The revealed bow shock effects on flow momentum can guide the assessment of cultivated forests in impeding geophysical flows. The revealed bow shocks and their ability to change the flow runout highlight the need to explore new models other than classical models based only on basal friction.

Convergence Analysis under Consistent Error Bounds

We introduce the notion of consistent error bound functions which provides a unifying framework for error bounds for multiple convex sets. This framework goes beyond the classical Lipschitzian and Hölderian error bounds and includes logarithmic and entropic error bounds found in the exponential cone. It also includes the error bounds obtainable under the theory of amenable cones. Our main result is that the convergence rate of several projection algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function. Another feature is the usage of Karamata theory and functions of regular variations which allows us to reason about convergence rates while bypassing certain complicated expressions. Finally, applications to conic feasibility problems are given and we show that a number of algorithms have convergence rates depending explicitly on the singularity degree of the problem.

Minimal positive realizations for linear systems with two-different-pole transfer functions

The minimal positive realization problem for linear systems is to find positive systems with a state space of minimal dimensions while keeping their transfer functions unchanged. Here the necessary and sufficient condition is established for the linear systems with third-order two-different-pole transfer functions to have a three-dimensional minimal positive realization by developing a new approach. The approach is to identify a canonical form of positive realizations with the help of polyhedral cone theory, and the identification consists of a sequential edge transformations with some invariant feature. The analyses are then successfully extended to the linear systems with general nth-order two-different-pole transfer functions. At the same time, the approach can also be used to generate infinitely many distinct minimal positive realizations for some linear systems.

A Novel Direct Optimization Framework for Hypersonic Waverider Inverse Design Methods

Waverider is a hypersonic vehicle that improves the lift-to-drag ratio using the shockwave attached to the leading edge of the lifting surface. Owing to its superior aerodynamic performance, it exhibits a viable external configuration in hypersonic flight conditions. Most of the existing studies on waverider employ the inverse design method to generate vehicle configuration. However, the waverider inverse design method exhibits two limitations; inaccurate definition of design space and unfeasible performance estimation during the design process. To address these issues, a novel framework to directly optimize the waverider is proposed in this paper. The osculating cone theory is adopted as a waverider inverse design method. A general methodology to define the design space is suggested by analyzing the design curves of the osculating cone theory. The performance of the waverider is estimated accurately and rapidly via combining a high-fidelity computational fluid dynamics solver and a surrogate model. A comparison study shows that the proposed direct optimization framework enables a more accurate design space and efficient performance estimation. The framework is applied to the multi-objective optimization problem, which maximizes internal volume and minimizes aerodynamic drag. Finally, general characteristics for waverider are presented by analyzing the optimized results with data mining methods such as K-means.

Conic Iterative Learning Control Using Distinct Data for Constrained Systems With State-Dependent Uncertainty

In batch processes, the ability to learn from previous process data results in high-value and batch-improved products. For batch processes with constraints and state-dependent uncertainty, this article presents a conic iterative learning control (ILC) approach, which uses cone theory to incorporate historical process data into optimization-based ILC design. The proposed conic ILC approach uses rank conditioning to select distinct data samples and conic mapping to map the data to the to-be-optimized control input variables, since adding all historical process data would be computationally intensive. Our method yields a tradeoff between learning ability from historical experience and computational efficiency from solving the optimization problem. Provable constraint satisfaction and robust stability are considered separately. To demonstrate the proven properties and effectiveness of the approach, we present a case study of the injection molding process.

An eigenvalue problem in fractional h-discrete calculus

In this paper, we prove existence of solutions for an eigenvalue problem in the fractional h-discrete calculus. Dirichlet type boundary conditions are considered. Several properties for Green’s function of the associated problem are proven. A fixed point theorem in Cone theory is a main tool to obtain sufficient conditions on upper and lower bounds for eigenvalues of the boundary value problem so that the existence of positive solutions are guaranteed.

Influence of thread parameters on the withdrawal capacity of wood screws to optimize the thread geometry

For wood screw connections, the withdrawal capacity is essential and is therefore an optimization target for wood screws. The withdrawal capacity is based on the composite action between the screw thread and the wood. Optimizing thread geometry requires knowledge of how thread parameters influence the withdrawal capacity. Up to now, this knowledge is largely unknown. Therefore, the objective is to determine the influence of thread height (1 mm; 1.48 mm), flank distance (3.04 mm; 6.08 mm), lead angle (6.8°; 13.6°), and thread angle (30°; 55°) on the withdrawal capacity. The influences are investigated for a tangential screw-in direction in spruce. For this, flat ribbed bars, so-called threaded test objects, are used. The effect on the withdrawal capacity is measured using an experimental setup based on test standard EN 1382 in two different planes of the flat thread to the wood fiber. 242 tests in RT plane and 286 in TL plane were analyzed with multifactorial analyses of variance. The thread height, flank distance, and thread angle show significant effects in both planes. A larger thread height, a smaller thread pitch, and a more acute thread angle increase the withdrawal capacity. Only in RT plane, a larger lead angle shows higher withdrawal capacity. The determined effects can be used to design the thread geometry of wood screws with higher withdrawal capacities. The accuracy of calculation models for the withdrawal capacity of wood screw connections can also be improved based on the findings. With the results, confirmation could be found for known models based on the theory of compression cones explaining the influences of the flank distance and thread angle.