Convex Cone Research Articles

Convex cones in mapping spaces between matrix algebras

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The duals of such convex cones can be characterized in terms of ampliation maps, which can also be used to characterize many notions from quantum information theory—such as separability, entanglement-breaking maps, Schmidt numbers, as well as decomposable maps and k-positive maps in functional analysis. In fact, such characterizations hold if and only if the involved cone is a one-sided mapping cone. Through this analysis, we obtain mapping properties for compositions of cones from which we also obtain several equivalent statements of the PPT (positive partial transpose) square conjecture.

Avoided quantum criticality in exact numerical simulations of a single disordered Weyl cone

Existing theoretical works differ on whether three-dimensional Dirac and Weyl semimetals are stable to a short-range-correlated random potential. Numerical evidence suggests the semimetal to be unstable, while some field-theoretic instanton calculations have found it to be stable. The differences go beyond method: the continuum field-theoretic works use a single, perfectly linear Weyl cone, while numerical works use tight-binding lattice models which inherently have band curvature and multiple Weyl cones. In this work, we bridge this gap by performing exact numerics on the same model used in analytic treatments, and we find that all phenomena associated with rare regions near the Weyl node energy found in lattice models persist in the continuum theory: The density of states is non-zero and exhibits an avoided transition. In addition to characterizing this transition, we find rare states and show that they have the expected behavior. The simulations utilize sparse matrix techniques with formally dense matrices; doing so allows us to reach Hilbert space sizes upwards of $10^7$ states, substantially larger than anything achieved before.

A Characterization of Nonnegativity Relative to Proper Cones

Let A be an m × n matrix with real entries. Given two proper cones K1 and K2 in ℝn and ℝm, respectively, we say that A is nonnegative if A(K1) ⊆ K2. A is said to be semipositive if there exists a $$x \in K_1^ \circ $$ such that $$Ax \in K_2^ \circ $$ . We prove that A is nonnegative if and only if A + B is semipositive for every semipositive matrix B. Applications of the above result are also brought out.

Sobolev inequalities with jointly concave weights on convex cones

Using optimal mass transport arguments, we prove weighted Sobolev inequalities of the form ∫ E | u ( x ) | q ω ( x ) d x 1 / q ⩽ K 0 ∫ E | ∇ u ( x ) | p σ ( x ) d x 1 / p , u ∈ C 0 ∞ ( R n ) , (WSI)where p ⩾ 1 and q > 0 is the corresponding Sobolev critical exponent. Here E ⊆ R n is an open convex cone, and ω , σ : E → ( 0 , ∞ ) are two homogeneous weights verifying a general concavity-type structural condition. The constant K 0 = K 0 ( n , p , q , ω , σ ) > 0 is given by an explicit formula. Under mild regularity assumptions on the weights, we also prove that K 0 is optimal in (WSI) if and only if ω and σ are equal up to a multiplicative factor. Several previously known results, including the cases for monomials and radial weights, are covered by our statement. Further examples and applications to partial differential equations are also provided.

Experimental and Numerical Simulation of Flow Pattern Evolution in a Flow Mixing Nozzle under a Moderate Inlet Flow Rate.

A flow mixing nozzle is a novel two-phase flow nozzle with a simple structure and has broad application prospects in the industrial field. However, the research on the working characteristics of this nozzle is mainly focused on a low inlet flow rate. There are few research studies on the working characteristics of the moderate and high flow rates which are more concerned in the industrial field, which limits the application of the flow mixing nozzle. In this paper, the experimental and simulation studies on the internal two-phase flow pattern and spray form of the flow mixing nozzle under a moderate inlet flow rate are carried out. The results show that the change in the two-phase inlet flow rate and the tube hole distance of the nozzle will affect the gas and liquid inertia force, thus affecting the working characteristics of the nozzle. The enhancement of gas inertia force will cause the flow pattern inside the nozzle to change to an unstable radially concave cone, and the spray form will be converted to atomization; the atomization effect of the liquid is better. The enhancement of the liquid inertial force will cause the flow pattern inside the nozzle to change to a radially convex cone, and the spray form will be converted to breakup; the atomization effect of the liquid is poor. The liquid surface tension maintains the existence of the cone inside the nozzle and plays a dominant role in jet atomization when the gas and liquid inertial forces are weak. In addition, the study also found that because of the compressibility of gas, the orifice has a reinforcing effect on gas inertia force.

Further characterizations of possibility-theoretical indices in fuzzy optimization

In this paper, we investigate six types of fuzzy relations based on Zadeh's possibility theory. Such relations were originally proposed by Dubois and Prade as comparison indices for fuzzy numbers and were extended by Inuiguchi, Ichihashi, and Kume to general cases. The fuzzy relations we consider are defined for fuzzy sets in a vector space preordered by a convex cone. We reveal the connection of the fuzzy relations with set relations, order-like crisp relations known in the area of set optimization. Specifically, we prove some equalities and equivalences involving the fuzzy relations and set relations under three different assumptions. We finally consider a fuzzy optimization problem in a general form and define its solution concepts. Applying the above equivalences to the problem, we obtain its relationship to certain set optimization problems.

A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles

We use convex polyhedral cones to study a large class of multivariate meromorphic germs, namely those with linear poles, which naturally arise in various contexts in mathematics and physics. We express such a germ as a sum of a holomorphic germ and a linear combination of special non-holomorphic germs called polar germs. In analyzing the supporting cones -- cones that reflect the pole structure of the polar germs -- we obtain a geometric criterion for the non-holomorphicity of linear combinations of polar germs. This yields the uniqueness of the above sum when required to be supported on a suitable family of cones and assigns a Laurent expansion to the germ. Laurent expansions provide various decompositions of such germs and thereby a uniformized proof of known results on decompositions of rational fractions. These Laurent expansions also yield new concepts on the space of such germs, all of which are independent of the choice of the specific Laurent expansion. These include a generalization of Jeffrey-Kirwan's residue, a filtered residue and a coproduct in the space of such germs. When applied to exponential sums on rational convex polyhedral cones, the filtered residue yields back exponential integrals.

Stochastic monotonicity of a distribution family associated with matrix projections and its application

This article defines a distribution family based on the distribution of a random variable associated with the projections of a normally distributed random matrix with unknown mean onto a linear space and onto a closed convex cone. The stochastic monotonicity property of the distributions in the family is established. This property finds the application in the proving the unbiasedness of a likelihood ratio test in a multivariate order restricted model.

Lifts of Non-Compact Convex Sets and Cone Factorizations

In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

Monotonicity Under Local Operations: Linear Entropic Formulas

All correlation measures, classical and quantum, must be monotonic under local operations. In this paper, we characterize monotonic formulas that are linear combinations of the von Neumann entropies associated with the quantum state of a physical system that has n parts. We show that these formulas form a polyhedral convex cone, which we call the monotonicity cone, and enumerate its facets. We illustrate its structure and prove that it is equivalent to the cone of monotonic formulas implied by strong subadditivity. We explicitly compute its extremal rays for n up to 5. We also consider the symmetric monotonicity cone, in which the formulas are required to be invariant under subsystem permutations. We describe this cone fully for all n. In addition, we show that these results hold even when states and operations are constrained to be classical.

Regularity conditions and Farkas-type results for systems with fractional functions

This paper deals with some new versions of Farkas-type results for a system involving cone convex constraint, a geometrical constraint as well as a fractional function. We first introduce some new notions of regularity conditions in terms of the epigraphs of the conjugate functions. By using these regularity conditions, we obtain some new Farkas-type results for this system using an approach based on the theory of conjugate duality for convex or DC optimization problems. Moreover, we also show that some recently obtained results in the literature can be rediscovered as special cases of our main results.

On limit behavior of a solution to boundary value problem in a plane sector

We study a certain boundary value problem in Sobolev‐Slobodetskii spaces with integral condition in a plane excluding a ray from origin. Using auxiliary problem in outside of a convex cone and the wave factorization concept, we construct a general solution and consider transfer to limit boundary value problem. It was shown that limit boundary value problem can be solvable only if the boundary function satisfies to a certain singular functional equation.

Principal eigenvalues for <i>k</i>-Hessian operators by maximum principle methods

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ of $\mathbb{R}^N$, a characterization of the principal eigenvalue associated to a $k$-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which <i>admissible viscosity supersolutions</i> obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $\Sigma_k \subset {\cal S}(N)$ which is an <i>elliptic set</i> in the sense of Krylov ^{[<span class="xref"><a href="#b23" ref-type="bibr">23</a></span>]} which corresponds to using $k$-convex functions as admissibility constraints in the formulation of viscosity subsolutions and supersolutions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property which results from the establishment of a global Hölder estimate for the unique $k$-convex solutions of the approximating equations.

Strictly Linear Light Cones in Long-Range Interacting Systems of Arbitrary Dimensions

In locally interacting quantum many-body systems, the velocity of information propagation is finitely bounded and a linear light cone can be defined. Outside the light cone, the amount of information rapidly decays with distance. When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists. Herein, we consider generic long-range interacting systems with decaying interactions, such as $R^{-\alpha}$ with distance $R$. We prove the existence of the linear light cone for $\alpha>2D+1$ ($D$: the spatial dimension), where we obtain the Lieb--Robinson bound as $\|[O_i(t),O_j]\|\lesssim{t}^{2D+1}(R-\bar{v}t)^{-\alpha}$ with $\bar{v}=\mathcal{O}(1)$ for two arbitrary operators $O_i$ and $O_j$ separated by a distance $R$. Moreover, we provide an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for $\alpha<2D+1$. In the regime of $\alpha>2D+1$, our result characterizes the best general constraints on the information spreading.

Hierarchy of Linear Light Cones with Long-Range Interactions

In quantum many-body systems with local interactions, quantum information and entanglement cannot spread outside of a linear light cone, which expands at an emergent velocity analogous to the speed of light. Local operations at sufficiently separated spacetime points approximately commute -- given a many-body state, $\mathcal{O}_x(t) \mathcal{O}_y |\psi\rangle \approx \mathcal{O}_y\mathcal{O}_x(t) |\psi\rangle$ with arbitrarily small errors -- so long as $|x-y|\gtrsim vt$, where $v$ is finite. Yet most non-relativistic physical systems realized in nature have long-range interactions: two degrees of freedom separated by a distance $r$ interact with potential energy $V(r) \propto 1/r^{\alpha}$. In systems with long-range interactions, we rigorously establish a hierarchy of linear light cones: at the same $\alpha$, some quantum information processing tasks are constrained by a linear light cone while others are not. In one spatial dimension, this linear light cone exists for every many-body state when $\alpha>3$ (Lieb-Robinson light cone); for a typical state chosen uniformly at random from the Hilbert space when $\alpha>\frac{5}{2}$ (Frobenius light cone); for every state of a non-interacting system when $\alpha>2$ (free light cone). These bounds apply to time-dependent systems and are optimal up to subalgebraic improvements. Our theorems regarding the Lieb-Robinson and free light cones -- and their tightness -- also generalize to arbitrary dimensions. We discuss the implications of our bounds on the growth of connected correlators and of topological order, the clustering of correlations in gapped systems, and the digital simulation of systems with long-range interactions. In addition, we show that universal quantum state transfer, as well as many-body quantum chaos, are bounded by the Frobenius light cone, and therefore are poorly constrained by all Lieb-Robinson bounds.

Order cones: a tool for deriving k-dimensional faces of cones of subfamilies of monotone games

In this paper we introduce the concept of order cone. This concept is inspired by the concept of order polytopes, a well-known object coming from Combinatorics with which order cones share many properties. Similarly to order polytopes, order cones are a special type of polyhedral cones whose geometrical structure depends on the properties of a partially ordered set (brief poset). This allows to study the geometrical properties of these cones in terms of the subjacent poset, a problem that is usually simpler to solve. Besides, for a given poset, the corresponding order polytope and order cone are deeply related. From the point of view of applicability, it can be seen that many cones appearing in the literature of monotone TU-games are order cones. Especially, it can be seen that the cones of monotone games with restricted cooperation are order cones, no matter the structure of the set of feasible coalitions and thus, they can be studied in a general way applying order cones.

On M-predecomposable sets

M-predecomposable set in $${\mathbf {R}}^n$$ is a convex set which is the sum of a compact convex set and a convex (not necessarily closed) cone. Complementing the existing results on M-predecomposable sets, we study their support and asymptotic properties. Also, we establish some properties of M-polyhedral sets, which are sums of compact convex sets and polyhedral cones.

Superbalance of holographic entropy inequalities

The domain of allowed von Neumann entropies of a holographic field theory carves out a polyhedral cone — the holographic entropy cone — in entropy space. Such polyhedral cones are characterized by their extreme rays. For an arbitrary number of parties, it is known that the so-called perfect tensors are extreme rays. In this work, we constrain the form of the remaining extreme rays by showing that they correspond to geometries with vanishing mutual information between any two parties, ensuring the absence of Bell pair type entanglement between them. This is tantamount to proving that besides subadditivity, all non-redundant holographic entropy inequalities are superbalanced, i.e. not only do UV divergences cancel in the inequality itself (assuming smooth entangling surfaces), but also in the purification thereof.

The Cheng-Yau metrics on regular convex cones as harmonic immersions into the symmetric space of positive definite real symmetric matrices

A Riemannian metric $g$ on a domain $\Omega$ in $ \mathbb{R}^n$ defines a map $F_g$ from $(\Omega,g)$ into the symmetric space of positive definite real symmetric $n \times n$ matrices $(\textrm{Sym}^+(n),h)$, where $h$ is the Cheng-Yau metric on $\textrm{Sym}^+(n)$. We show that the map $F_g$ is a harmonic immersion if $\Omega$ is a regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$. We also prove that the map $F_g$ is totally geodesic if $\Omega$ is a homogeneous self-dual regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$.

Ordered Gyrovector Spaces

The well-known construction scheme to define a partial order on a vector space is to use a proper convex cone. Applying this idea to the gyrovector space we construct the partial order, called a gyro-order. We also give several inequalities of gyrolines and cogyrolines in terms of the gyro-order.