Convex Cone Research Articles

Nonsymmorphic symmetry protected spin-orbit semi-Dirac fermions in two dimensions

Semi-Dirac fermions (SDFs) in two-dimensional (2D) systems, which simultaneously exhibit linear and quadratic dispersion around them, are a bridge linking linear and quadratic Dirac cones. However, the so-called SDFs in the reported 2D materials are doubly degenerate (namely semi-Weyl fermions), and are vulnerable against spin-orbital coupling (SOC). Here, we propose a 2D SOC-robust SDF which arises from nonsymmorphic symmetries. Unlike the known 2D SDFs, due to the presence of inversion and time reversal symmetries ($\mathcal{I}\mathcal{T}$), each crossing band is doubly degenerate, making the SDF of fourfold degeneracy. By high throughput screening, we find that 26 candidate 2D materials (such as ${\mathrm{CuCN}}_{2}$, ${\mathrm{CaI}}_{2}$, TlF) can hold SDFs, which belong to layer groups 40 ($pmam$), 43 ($pbaa$), and 45 ($pbma$), respectively. Furthermore, the symmetry protection mechanism, associated Fermi arc edge states, the catalytic performance, and the topological phase transitions under symmetry breaking of SDFs are revealed, providing a way for understanding 2D SOC-robust SDFs. Overall, our work proposes a class of topological phases, and provides a platform to study their fascinating physical effects.

Tropical fans and normal complexes: Putting the “volume” back in “volume polynomials”

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.

A Mean Value Theorem for Tangentially Convex Functions

The main result is an equality type mean value theorem for tangentially convex functions in terms of tangential subdifferentials, which generalizes the classical one for differentiable functions, as well as Wegge theorem for convex functions. The new mean value theorem is then applied, analogously to what is done in the classical case, to characterize, in the tangentially convex context, Lipschitz functions, increasingness with respect to the ordering induced by a closed convex cone, convexity, and quasiconvexity.

Exposing the threshold structure of loop integrals

The understanding of the physical laws determining the infrared behavior of amplitudes is a longstanding and topical problem. In this paper, we show that energy conservation alone implies strong constraints on the threshold singularity structure of Feynman diagrams. In particular, we show that it implies a representation of loop integrals in terms of Fourier transforms of nonsimplicial convex cones. We then engineer a triangulation that has a direct diagrammatic interpretation in terms of a straightforward edge-contraction operation. We use it to develop an algorithmic procedure that performs the Fourier integrations in closed form, yielding the novel cross-free family three-dimensional representation of loop integrals. Its singularity structure is entirely and elegantly expressed in terms of the graph-theoretic notions of connectedness and crossing. These results can be used to study the Kinoshita-Lee-Nauenberg cancellation mechanism, numerically evaluate loop integrals and to simplify threshold regularization procedures.

Mathematical Reflections on Locality

Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, we review some features of the notion of locality arising in physics and mathematics. We encode these in locality relations, given by symmetric binary relations, and locality morphisms, namely maps that factorise on products of pairs in the graph of such locality relations. This factorisation is a key property in the context of renormalisation, as illustrated on the factorisation of an exponential sum on convex cones, discussed at the end of the paper. The subject of locality is so vast and the issues it raises are so subtle, that this brief and modest presentation can only offer a small glimpse into this fascinating topic.

On a faithful representation of Sturmian morphisms

The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we define a faithful representation by (3×3)-matrices with integer entries. We find three convex cones in R3 and show that a matrix R∈Sl(Z,3) is a matrix representing a Sturmian morphism if the three cones are invariant under multiplication by R or R−1. This property offers a new tool to study Sturmian sequences. We provide alternative proofs of four known results on Sturmian sequences fixed by a primitive morphism and a new result concerning the square root of a Sturmian sequence.

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.

Applications of Strassen’s theorem and Choquet theory to optimal transport problems, to uniformly convex functions and to uniformly smooth functions

We provide a unifying interpretation of various optimal transport problems as a minimisation of a linear functional over the set of all Choquet representations of a given pair of probability measures ordered with respect to a certain convex cone of functions. This allows us to provide novel proofs of duality formulae. Among our tools is Strassen’s theorem.We provide new formulations of the primal and the dual problem for martingale optimal transport employing a novel representation of the set of extreme points of probability measures in convex order on Euclidean space. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions.We introduce a notion of martingale triangle inequality. We show that Kantorovich–Rubinstein duality bears an analogy in the martingale setting employing the cost functions that satisfy the inequality.

On the foundations and extremal structure of the holographic entropy cone

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. We also report on some partial results for $6$ terminals. Some interesting open problems are stated throughout.

Energy-Efficient Beamforming Design for User-Centric Networks With Full-Duplex Wireless Fronthaul

In this paper, we investigate the joint beamforming design for access and fronthaul links in a user-centric network with full-duplex fronthaul. We formulate an optimization problem to maximize the network energy efficiency while guaranteeing fronthaul rate requirements. Particularly, the power consumed by self-interference cancellation at full-duplex access points (APs) is taken into consideration to comprehensively reflect the effect of full-duplex technology. Due to the non-convexity and the nonlinear fractional form of the formulated problem, we resort to the successive convex approximation method. The original problem is approximated as a convex second-order cone program. We then extend the study to the scenario of imperfect and partial channel state information (CSI). To avoid the intractable expression of the achievable rate in this case, we derive its lower bound for further beamforming design. Simulation results demonstrate the superiority of the proposed algorithm in terms of energy efficiency and power consumption under both CSI scenarios. For our considered full-duplex fronthaul networks, moderate maximum transmit power of macro base station (MBS) and APs is sufficient to obtain desired energy efficiency performance. Moreover, the lower bound is verified to be tight at the low-maximum power region.

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.

An algebraic approach to the variants of convexity for soft expert approximate function with intuitionistic fuzzy setting

A new area of research called intuitionistic fuzzy soft expert set is expected to overcome the drawbacks of an intuitionistic fuzzy soft set in terms of eligibility for soft expert-argument approximate function. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert-argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, ( α , v ) -convexity(concave) sets, strongly ( α , v ) -convexity (concave) sets, strictly ( α , v ) -convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and results on the basis of these specified notions. Additionally, by using a theoretic cum analytical approach, various elements of computational geometry, such as convex hull and convex cone, are theorized and some pertinent results are generalized.

Applications of Fell Topology to Closed Set-Valued Optimizations in Partially Ordered First Countable Topological Vector Spaces

Let (X, ) be a topological vector space equipped with a partial order which is induced by a nonempty pointed closed and convex cone K in X. Let (X) = 2 X be the power set of X and (X) = 2 X \\{}. In this paper, we consider the so called upward preordering on (X), which has been used by many authors (see [47, 9, 1114, 1819]). We first prove some properties of this ordering relation Let (X) denote the collection of all -closed subsets of X and (X) = (X)\\{}, which is equipped with the Fell topology Let C be a nonempty subset of X and let F: C (X) be a closed set-valued mapping. By applying the Fell topology on (X) and the Fan-KKM Theorem, we prove some existence theorems for some -minimization and -maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in

Strategies for single-shot discrimination of process matrices

The topic of causality has recently gained traction quantum information research. This work examines the problem of single-shot discrimination between process matrices which are an universal method defining a causal structure. We provide an exact expression for the optimal probability of correct distinction. In addition, we present an alternative way to achieve this expression by using the convex cone structure theory. We also express the discrimination task as semidefinite programming. Due to that, we have created the SDP calculating the distance between process matrices and we quantify it in terms of the trace norm. As a valuable by-product, the program finds an optimal realization of the discrimination task. We also find two classes of process matrices which can be distinguished perfectly. Our main result, however, is a consideration of the discrimination task for process matrices corresponding to quantum combs. We study which strategy, adaptive or non-signalling, should be used during the discrimination task. We proved that no matter which strategy you choose, the probability of distinguishing two process matrices being a quantum comb is the same.

Origin of neutrino masses on the convex cone of positivity bounds

We exhibit the geometric structure of the convex cone in the linear space of the Wilson coefficients for the dimension-eight operators involving the left-handed lepton doublet $L$ and the Higgs doublet $H$ in the Standard Model effective field theory (SMEFT). The boundary of the convex cone gives rise to the positivity bounds on the Wilson coefficients, while the extremal ray corresponds to the unique particle state in the theory of ultraviolet completion. Among three types of canonical seesaw models for neutrino masses, we discover that only right-handed neutrinos in the type-I seesaw model show up as one of the extremal rays, whereas the heavy particles in the type-II and type-III seesaw models live inside the cone. The experimental determination of the relevant Wilson coefficients close to the extremal ray of the type-I seesaw model will unambiguously pin down or rule out the latter as the origin of neutrino masses. This discovery offers a novel way to distinguish the most popular seesaw model from others, and it also strengthens the SMEFT as an especially powerful tool to probe new physics beyond the Standard Model.

Decomposability of multiparameter car flows

AbstractLet P be a closed convex cone in $\mathbb{R}^d$ which is assumed to be spanning $\mathbb{R}^d$ and contains no line. In this article, we consider a family of CAR flows over P and study the decomposability of the associated product systems. We establish a necessary and sufficient condition for CAR flow to be decomposable. As a consequence, we show that there are uncountable many CAR flows which are cocycle conjugate to the corresponding CCR flows.

Controllability Properties of Solar Sails

Trajectory design and stationkeeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when reemitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, the force generated by the solar radiation pressure is contained in a pointed convex cone of revolution with the axis in the sun–satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective because it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.

Geometric Duality Results and Approximation Algorithms for Convex Vector Optimization Problems

We study geometric duality for convex vector optimization problems. For a primal problem with a -dimensional objective space, we formulate a dual problem with a -dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter, and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in such a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures.

Action-angle coordinates on coadjoint orbits and multiplicity free spaces from partial tropicalization

Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups K=Un,SOn. Extending these results to groups of other types is one of the goals of this paper.Partial tropicalizations are Poisson spaces with constant Poisson bracket. They provide a bridge between dual spaces of Lie algebras Lie(K)⁎ with linear Poisson brackets and polyhedral cones which parametrize the canonical bases of irreducible modules of G=KC.We generalize the construction of partial tropicalizations to allow for arbitrary cluster charts, and apply it to questions in symplectic geometry. For each regular coadjoint orbit of a compact group K, we construct an exhaustion by symplectic embeddings of toric domains. As a by product we are able to complete the proof of a long-standing conjecture due to Karshon and Tolman about the Gromov width of regular coadjoint orbits. We also construct an exhaustion by symplectic embeddings of toric domains for multiplicity free K-spaces.An essential tool in our study is the dual Poisson-Lie group K⁎ equipped with the Berenstein-Kazhdan potential Φ. Partial tropicalizations arise as tropical limits of K⁎, and the potential Φ defines the range of action variables of a Gelfand-Zeitlin type integrable system. Our results give rise to new questions and conjectures about the Poisson structure of K⁎ and Ginzburg-Weinstein Poisson isomorphisms Lie(K)⁎→K⁎.

State-Space Region-Invariance of Discrete-Time Control Systems

In the paper, the so called region-invariance properties of discrete-time control systems are considered. An already known notion of positivity of control systems has been generalized (using a kind of geometric insight instead of a pure algebraic approach) to the general nonlinear discrete-time control systems, general regions in state-space, and controls from polyhedral cones in input-space. The class of such control systems has been characterized. Numerous numerical examples illustrating individual cases under consideration are presented in the paper.