Symmetric Cone Research Articles

Implications of Asymmetric Loss Cone Distribution on Whistler‐Driven Electron Precipitation at Mercury

AbstractMercury has a large loss cone difference in its two hemispheres due to the northward shifted magnetic dipole. The precipitation difference of energetic electrons in both hemispheres is poorly understood. We show that the northern precipitation is 2.5‐times higher than for a symmetric loss cone due to the effects of the enhanced whistler instability at the southern hemisphere with the larger loss cone. Simulations including nonlinear pitch angle scattering by the whistler‐mode waves show rapid (tens of milliseconds) electron flux modulation related to the wave subpacket structures by repeated interactions within a discrete wave element. The difference in the nonlinear whistler instability in the two hemispheres should enhance the electron precipitation, which, along with the direct impact effects of solar wind, contributes to Mercury's surface–magnetosphere coupling. Electrons hitting the planet's surface may be a possible factor in the formation of water through the formation of hydroxyl groups.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Jordan automorphisms and derivatives of symmetric cones

Hyperbolicity cones, and in particular symmetric cones, are of great interest in optimization. Renegar showed that every hyperbolicity cone has a family of derivative cones that approximate it. Ito and Lourenço found the automorphisms of those derivatives when the original cone is generated by rank-one elements, as symmetric cones happen to be. We show that the derivative automorphisms of a symmetric cone are closely related to the automorphisms of its associated Euclidean Jordan algebra. In the process, we find the automorphism group of the quaternion positive-semidefinite cone and list explicitly the Jordan-automorphisms of the quaternion Hermitian matrices. We also address the path-connectedness of the simple Euclidean Jordan-automorphism groups.

A coordinated restoration method of three-phase AC unbalanced distribution network with DC connections and mobile energy storage systems

In the rapidly changing domain of hybrid AC/DC urban distribution networks, this research unveils a groundbreaking method for the restoration of three-phase unbalanced systems by astutely harnessing the unique potential of DC line interconnections. At the heart of this innovation lies the synthesis of symmetrical Second Order Cone Programming (SOCP) with a sophisticated topology search technique, a union that offers a precise depiction of complex three-phase power flow with network restoration while simultaneously accelerating computational processes. Building upon this foundation, our approach places significant emphasis on the utilization of adaptable DC power control, coupled with the optimal deployment of mobile energy storage systems (MESSs), to ensure a harmonized power balance during critical interruptions. These strategies converge to prioritize the restoration of vital loads, especially those with high weighting factors, thereby significantly augmenting the network’s resilience, particularly in contexts vulnerable to disasters. The corroborative numerical results, as delineated in our study, highlight the distinct advantage and effectiveness of our methodology over prevailing practices in fortifying grid resilience against serious adversities.

Maximum principles and consequences for γ$\gamma$‐translators in Rn+1${\mathbb {R}}^{n+1}$

AbstractIn this paper, we obtain several properties of translating solitons for a general class of extrinsic geometric curvature flow, where the deformation speed is given by a homogeneous smooth symmetric positive function defined in a symmetric open cone. The main result of this paper is about the uniqueness of ‐translators in the class of complete graphs defined on a ball.

Hankel transform, [formula omitted]-Bessel functions and zeta distributions in the Dunkl setting

We study analytic properties of a Hankel transform for the type A Dunkl setting with arbitrary multiplicity parameter k≥0 which goes back to Baker and Forrester and, in an earlier symmetrized version, to Macdonald. Moreover, we introduce a Dunkl analogue of the Bessel functions and K-Bessel functions, generalizing those of a symmetric cone which occur when the multiplicity parameter k is chosen to be twice the Peirce dimension constant of the cone. In particular, we show that our K-Bessel function is a solution of a type B system of Dunkl operators which is a generalization of the Bessel system on a symmetric cone. Furthermore, we define zeta distributions and prove a functional equation relating them with their type B Dunkl transform. In the case of symmetric cones the aforementioned functional equation reduces to the known functional equation between zeta distributions and their Fourier transform. For the proof of this functional equation, the K-Bessel function and its analytic properties turn out to be crucial. Finally, we examine which of the zeta distributions are given by a positive measure.

Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming

Optimality Conditions for Nonlinear Second-Order Cone Programming and Symmetric Cone Programming

Horofunction compactifications of symmetric cones under Finsler distances

In this paper we consider symmetric cones equipped with invariant Finsler distances, namely the Thompson distance and the Hilbert distance. We give a complete characterisation of the horofunctions of the symmetric cone \(A_+^\circ\) under the Thompson distance and establish a correspondence between the horofunction compactification of \(A_+^\circ\) and the horofunction compactification of the normed space in the tangent bundle. More precisely, we show that the exponential map extends as a homeomorphism between the horofunction compactification of the normed space in the tangent bundle, which is a JB-algebra, and the horofunction compactification of \(A_+^\circ\). Analogues results are established for the Hilbert distance on the projective symmetric cone \(PA_+^\circ\). The analysis yields a concrete description of the horofunction compactifications of these spaces in terms of the facial structure of the closed unit ball of the dual norm of the norm in the tangent space.

A reflection formula for the Gaussian hypergeometric function of matrix argument

We obtain a reflection formula for the Gaussian hypergeometric function of real symmetric matrix argument. We also show that this result extends to the Gaussian hypergeometric function defined over the symmetric cones, and even to generalizations of the Gaussian hypergeometric function defined in terms of series of Jack polynomials. Finally, we obtain a quadratic transformation formula for the Gaussian hypergeometric function of Hermitian 2×2 matrix argument.

Conical diffraction in atomic vapor: Mathematical models and numerical calculations

This study investigated the conical diffraction characteristics in the honeycomb lattice of a Λ-type three-level energy system for rubidium atomic configuration. Its band structure incorporates Dirac cones in accordance with the plane wave expansion method. We determined that the rotation angle θ of the three vectors strongly impacted the cone diffraction profile. For θ=0° and 60°, in the reciprocal lattice space, the Dirac cone states were excited, and symmetric cone diffraction was supported. When 0<θ<60°, an asymmetric cone diffraction with a dark notch appeared on the outer edge of the conical ring. The results indicate that the frequency detuning of the coupling field and high-order nonlinear susceptibilities in the medium are effective and concise approaches to manipulate the occurrence of cone diffraction. The findings of this study enhance the understanding of the cone diffraction phenomena and broaden the practical applicability of atomic configuration.

High-sensitivity electrocardiogram sensor based on Fano resonance in a double-stub-assisted plasmonic micro-ring resonator

Detecting Electrocardiogram (ECG) signals remains a challenge due to its low signal amplitude and low frequency characteristic. In this paper, we suggest a solution to ECG sensor based on Fano resonance of surface plasma wave (SPW) in a Double-Stub-Assisted plasmonic micro-ring resonator (DS-PMRR) structure by exploiting the electro-optic (EO) effect of EO polymer. A theoretical model of Fano resonance of SPW in DS-PMRR is established to analyze the sensitivity of ECG sensor. A moderately symmetrical cone mode converter is designed to achieve the efficient coupling between silicon on insulator (SOI) waveguide and metal–insulator–metal (MIM) waveguide. The whole structure with an active zone as small as 5 μm × 8.4 μm × 0.2 μm is analyzed by finite-element method (FEM) and a sensitivity of 11740 pm/V is obtained. In combination with a sub-picometer optical spectrum analyzer, the proposed sensor can detect ECG signals with the magnitude on the order of 10 μv.

Investigation on the Mechanical Properties and Shape Memory Effect of Landing Buffer Structure Based on NiTi Alloy Printing

With the deepening of human research on deep space exploration, our research on the soft landing methods of landers has gradually deepened. Adding a buffer and energy-absorbing structure to the leg structure of the lander has become an effective design solution. Based on the energy-absorbing structure of the leg of the interstellar lander, this paper studies the appearance characteristics of the predatory feet of the Odontodactylus scyllarus. The predatory feet of the Odontodactylus scyllarus can not only hit the prey highly when preying, but also can easily withstand the huge counter-impact force. The predatory feet structure of the Odontodactylus scyllarus, like a symmetrical cone, shows excellent rigidity and energy absorption capacity. Inspired by this discovery, we used SLM technology to design and manufacture two nickel-titanium samples, which respectively show high elasticity, shape memory, and get better energy absorption capacity. This research provides an effective way to design and manufacture high-mechanical energy-absorbing buffer structures using bionic 3D printing technology and nickel-titanium alloys.

A Smoothing Method for Sparse Programs by Symmetric Cone Constrained Generalized Equations

In this paper, we consider a sparse program with symmetric cone constrained parameterized generalized equations (SPSCC). Such a problem is a symmetric cone analogue with vector optimization, and we aim to provide a smoothing framework for dealing with SPSCC that includes classical complementarity problems with the nonnegative cone, the semidefinite cone and the second-order cone. An effective approximation is given and we focus on solving the perturbation problem. The necessary optimality conditions, which are reformulated as a system of nonsmooth equations, and the second-order sufficient conditions are proposed. Under mild conditions, a smoothing Newton approach is used to solve these nonsmooth equations. Under second-order sufficient conditions, strong BD-regularity at a solution point can be satisfied. An inverse linear program is provided and discussed as an illustrative example, which verified the efficiency of the proposed algorithm.

Analogues of Poisson-type limit theorems in discrete bm-Fock spaces

In this paper, we present analogues of the Poisson limit distribution for the noncommutative bm-independence, which is associated with several positive symmetric cones. We construct related discrete Fock spaces with creation, annihilation and conservation operators, and prove Poisson type limit theorems for them. Properties of the positive cones, in particular the volume characteristic property they enjoy, and the combinatorics of labeled noncrossing partitions, play crucial role in these considerations.

Approximation hierarchies for copositive cone over symmetric cone and their comparison

We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (Structured semidefinite programs and semialgebraic geometry methods in Robustness and optimization, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) or Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, https://doi.org/10.1137/03060151X, 2006) and Lasserre (Math Program 144:265–276, https://doi.org/10.1007/s10107-013-0632-5, 2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) and Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.

A new extension of Chubanov’s method to symmetric cones

We propose a new variant of Chubanov’s method for solving the feasibility problem over the symmetric cone by extending Roos’s method (Optim Methods Softw 33(1):26–44, 2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (1) equivalent to that of Roos’s original method (2018) and superior to that of Lourenço et al.’s method (Math Program 173(1–2):117–149, 2019) when the symmetric cone is the nonnegative orthant, (2) superior to that of Lourenço et al.’s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (3) equivalent to that of Lourenço et al.’s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (4) superior to that of Pena and Soheili’s method (Math Program 166(1–2):87–111, 2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili’s method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating strongly (but ill-conditioned) feasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

A Geodesic Interior-Point Method for Linear Optimization over Symmetric Cones

A Geodesic Interior-Point Method for Linear Optimization over Symmetric Cones

Interior-point algorithm for symmetric cone horizontal linear complementarity problems based on a new class of algebraically equivalent transformations

AbstractWe generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) to $$P_*(\kappa )$$ P ∗ ( κ ) -horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illés T, Rigó PR, Török R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.

Design of a optimization algorithm for binary classification

In the present work, the design of a system to classify data is carried out, using the Scrum methodology. The validation was carried out by expert judgment, having favorable results in terms of different criteria such as; integrity, ease of use, innovation, and scalability. Regarding the development of the functional elements of the system, it was obtained; he developed the architecture of the system, the database, and the prototypes, among other points considered. From the implementation of the system, the equation of a classifying plane in three-dimensional space will be obtained, as well as the number of internal iterations that the algorithm develops, the estimated execution time, and the graph of the plane. This system is based on a recently introduced symmetric cone proximal multiplier algorithm to solve separable optimization problems, this algorithm made an application for classification-related support vector machines.

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.