Cone Theory Research Articles

Hitchin systems for invariant and anti-invariant vector bundles

Given a smooth projective complex curve X X with an involution σ \sigma , we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over X X under σ \sigma . Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.

An underactuated parallel-link gripper for a multicopter capable of plane perching

The need for a perching robot is increasing in the field of rescue and transportation. Accordingly studies on perching an object by attaching a robot arm to a perching robot have been conducted. However, almost all the studies related to perching have been conducted using an actuated or electric device. However, perching by using an electric device has several disadvantages, such as additional power consumption and an increase in the mass of the multicopter used to load the electric source. Instead of using an electric device, perching by using an underactuated gripper can effectively avoid these disadvantages. Accordingly, we developed an underactuated passive gripper that has the advantage of nonconsumption of electric power for perching. A method to confirm the available range for stable perching is one of the problems of using an underactuated passive gripper. Therefore, in this study, we analyze a multicopter carrying an underactuated parallel-link passive gripper for available plane perching. To enable perching on planes with different thicknesses and being embedded at different depths, we summarize the available perching range and limitations based on the friction cone theory. Our conclusion is supported by both theoretical and experimental results.

Dirac cones and higher-order topology in quasi-continuous media

We consider the Dirac cones and higher-order topological phases in quasi-continuous media of classical waves (e.g., photonic and sonic crystals). Using sonic crystals as prototype examples, we revisit some of the known systems in the study of topological acoustics. We show the emergence of various Dirac cones and higher-order topological band gaps in the same framework by tuning the geometry of the system. We provide a pedagogical review of the underlying physics and methodology via the bulk-edge-corner correspondence, symmetry-based indicators, Wannier representations, filling anomaly, and fractional corner charges. In particular, the theory of the Dirac cones and the higher-order topology are put in the same framework. These examples and the underlying physics principles can be inspiring and useful in the future study of higher-order topological metamaterials.

Special Vinberg cones and the entropy of BPS extremal black holes

We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged N = 2 D = 4 supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg’s theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of N = 2 supergravity with homogeneous scalar manifold.

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; \U0001d524) and {a}_{ji}={a}_{ij}^{ast }=mathfrak{g}left({a}_{ij},cdot right)in {V}_{ij}^{ast } belongs to the dual space {V}_{ij}^{ast }. The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric \U0001d524S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone \U0001d4b1 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone \U0001d4b12 ⊂ ℋ2(V ) to the special rank 3 case.

Existence and uniqueness of fixed points for monotone operators in partially ordered Banach spaces and applications

In this paper, we employ partial order method, cone theory, and the techniques of measure of weak noncompactness to prove several new theorems on the existence and the uniqueness of fixed points or coupled fixed points for operators satisfying some monotonicity assumptions. Our conclusions generalize and improve several well-known results. As an application, we investigate the existence of a unique solution for a class nonlinear second-order ordinary differential equations. We also discuss the existence of a unique solution for a system of integral equations.

PROPERTIES AND UNIQUE POSITIVE SOLUTION FOR FRACTIONAL BOUNDARY VALUE PROBLEM WITH TWO PARAMETERS ON THE HALF-LINE

<abstract> Based on the theory of cone and operators, this paper concerned with the existence of unique positive solution for a class of nonlinear fractional boundary value problem with two parameters (one is called an eigenvalue parameter and another is a disturbance parameter) on the half-line. More important, the solutions dependence on two parameters was discussed, which shows that different parameters have different effects on the properties of positive solutions, and the results reflect an interesting fact different from our inference. Some examples are given to illustrate the main results. </abstract>

A Cone-Dominance Approach for Discrete Alternative Multiple Criteria Problems with Indifference Regions

In this paper, we address the problem of solving discrete alternative multiple criteria decision making (MCDM) problems where some or all of the criteria might have indifference regions. We utilize the classical cone-dominance approach and significantly extend the associated theory. We then develop a convergent solution method based on cone-dominance for achieving the most preferred choice. The convergent method utilizes pairwise comparisons of the alternatives by the decision maker (DM) to eliminate inferior or dominated alternatives and to arrive at the optimum. The stated theoretical development significantly strengthens the theory of cones and presents a streamlined approach for solving the stated MCDM problems. We present a numerical example to illustrate the application of the method in finance. We also present a simulation study, evaluating the performance of the method on several hundred randomly generated test problems. Results of the simulation study are analyzed to assess the possible effects of the presence of indifference regions on the required number of pairwise comparisons to reach the optimal choice.

Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

Existence of positive solutions for the nonlinear algebraic system x=lambda GF ( x ) has been extensively studied when the ntimes n coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

Positive Solutions for Two-Point Boundary Value Problems for Fourth-Order Differential Equations with Fully Nonlinear Terms

In this paper, we consider the existence of positive solutions for the fully fourth-order boundary value problemu4t=ft,ut,u′t,u″t,u‴t, 0≤t≤1,u0=u1=u″0=u″1=0, wheref:0,1×0,+∞×−∞,+∞×−∞,0×−∞,+∞⟶0,+∞is continuous. This equation can simulate the deformation of an elastic beam simply supported at both ends in a balanced state. By using the fixed-point index theory and the cone theory, we discuss the existence of positive solutions of the fully fourth-order boundary value problem. We transform the fourth-order differential equation into a second-order differential equation by order reduction method. And then, we examine the spectral radius of linear operators and the equivalent norm on continuous space. After that, we obtain the existence of positive solutions of such BVP.

Rocking Response of Shallow Foundations in Time Domain Using Cone Model Theory

ABSTRACT To assess the dynamic stiffness of shallow foundations, as an alternative to the existing rigorous approaches, it is convenient to idealize the soil as some semifinite truncated cones representing the infinite nature of the supporting soil. The one-dimensional wave equation for cones which is usually defined in frequency domain has been extended to be solved directly in the time domain. The recently streamlined strength of materials using cones has been developed in the time domain to calculate the response of shallow foundations on a layered half-space for translational degrees of freedom, while the wave pattern in the time domain is not studied for a rocking degree of freedom. The concept of echoes which converts the response of the footing on homogenous half-space to that of the footing on layered site is still applicable when the layered system excited by rocking moments. The echo constants are derived in the current research for rotational degrees of freedom in either the flexibility or stiffness formulation. In this paper, using the theory of cones, equations for direct analysis of soil-structure interaction in the time domain are obtained for the rocking degrees of freedom. The results, calculated exclusively in the familiar time domain, are in remarkable agreement with the results obtained in the frequency domain.

Cosmic Spin and Mass Evolution of Black Holes and Its Impact

Abstract We build an evolution model of the central black hole that depends on the processes of gas accretion, the capture of stars, mergers, and electromagnetic torque. In the case of gas accretion in the presence of cooling sources, the flow is momentum driven, after which the black hole reaches a saturated mass; subsequently, it grows only by stellar capture and mergers. We model the evolution of the mass and spin with the initial seed mass and spin in ΛCDM cosmology. For stellar capture, we have assumed a power-law density profile for the stellar cusp in a framework of relativistic loss cone theory that includes the effects of black hole spin, Carter’s constant, loss cone angular momentum, and capture radius. Based on this, the predicted capture rates of 10−5 to 10−6 yr−1 are closer to the observed range. We have considered the merger activity to be effective for z ≲ 4, and we self-consistently include the Blandford–Znajek torque. We calculate these effects on the black hole growth individually and in combination, for deriving the evolution. Before saturation, accretion dominates the black hole growth (∼95% of the final mass), and subsequently stellar capture and mergers take over with roughly equal contributions. The simulations of the evolution of the M •–σ relation using these effects are consistent with available observations. We run our model backward in time and retrodict the parameters at formation. Our model will provide useful inputs for building demographics of the black holes and in formation scenarios involving stellar capture.

When a maximal angle among cones is nonobtuse

Principal angles between linear subspaces have been studied for their application to statistics, numerical linear algebra, and other areas. In 2005, Iusem and Seeger defined critical angles within a single convex cone as an extension of antipodality in a compact set. Then, in 2016, Seeger and Sossa extended that notion to two cones. This was motivated in part by an application to regression analysis, but also allows their cone theory to encompass linear subspaces which are themselves convex cones. One obstacle to computing the maximal critical angle between cones is that, in general, the maximum will not occur at a pair of generators of the cones. We show that in the special case where the maximal angle between the cones is nonobtuse, it does suffice to check only the generators. This special case can be checked at essentially no extra cost, and we incorporate that information into an improved algorithm to find the maximal angle.

Enhancement of droplet ejection from molten and liquid plasma-facing surfaces by the electric field of the sheath**Substantial portions of this paper are adapted from section 4.1 of the first author’s recent PhD thesis.

Maintaining the stability of a liquid surface in contact with a plasma is of crucial importance in a range of industrial and fusion applications. The most fundamental feature of a plasma-surface interaction, the formation of a highly-charged sheath region, has been neglected from the majority of previous studies of plasma-liquid interactions. This paper considers the effect of the electric field of the sheath on the ejection of micron-scale droplets from bubbles bursting at the liquid surface. A numerical simulation method, based on the ideal electrohydrodynamic model, is introduced and validated against the well-known Taylor cone theory. This model is then used to include the electrical effects of the sheath in simulations of bubble bursting events at a plasma-liquid interface. The results show a significant enhancement in droplet ejection at modest electric fields of between 10% and 20% of the critical field strength required for a solely electrohydrodynamic instability. This finding is in good qualitative agreement with experimental observations and its importance in a wide range of fusion and atmospheric-pressure plasma-liquid interactions is discussed. The inclusion of sheath physics in future studies of plasma-liquid interactions is strongly advocated.

Water entry behaviours of hemisphere-head projectiles with high velocity

The water-entry behaviours of projectiles with hemisphere-head were studied experimentally and theoretically, focusing on projectile dynamics and drag coefficient. Based on equivalent cone theory proposed by Baldwin, the equation of drag coefficient was developed to describe the resistance of hemispherical head projectiles from the time when the projectile contacts the water to the time the hemispherical head is submerged. A set of experimental apparatus was set up to record the water-entry process by high-speed photography, the variation of resistance during water-entry was obtained experimentally. Compared the acceleration calculated by conventional method, new method and experimental results, the new method is more consistent with the experimental value.

Signed and sign-changing solutions of bi-nonlocal fourth order elliptic problem

We study a class of nonlocal fourth order elliptic problem which is seldom studied because of the presence of the biharmonic operator and binonlocal terms. We are interested in the existence of signed and sign-changing solutions. Our approach is based on variational invariant sets of descending flow and cone theory. Our results extend and improve some recent work.

Fixed-Point Theorems for Systems of Operator Equations and Their Applications to the Fractional Differential Equations

We study the existence and uniqueness of positive solution for a class of nonlinear binary operator equations systems by means of the cone theory and monotone iterative technique, under more general conditions. Also, we give the iterative sequence of the solution and the error estimation of the system. Moreover, we use this new result to study the existence and uniqueness of the solutions for fractional differential equations systems involving integral boundary value conditions in ordered Banach spaces as an application. The results obtained in this paper are more general than many previous results and complement them.

Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces

In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces are established under more general conditions by constructing a special Banach space and using cone theory and the Banach contraction mapping principle. The results obtained herein improve and generalize some well-known results.

Volterra operators in the edge-calculus

We study the Volterra property of a class of anisotropic pseudo-differential operators on $$\mathbb {R}\times B$$ for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable $$\tau $$ of t extends to symbols with respect to $$\tau $$ to the lower complex v half-plane. The resulting space of Volterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph.D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113–137, 1997; Osaka J Math 37:221–260, 2000).

Existence and uniqueness of solutions for systems of fractional differential equations with Riemann\u2013Stieltjes integral boundary condition

In this article, we first establish an existence and uniqueness result for a class of systems of nonlinear operator equations under more general conditions by means of the cone theory and monotone iterative technique. Furthermore, the iterative sequence of the solution and the error estimation of the system are given. Then we use this new result to study the existence and uniqueness of the solution for boundary value problems of systems of fractional differential equations with a Riemann–Stieltjes integral boundary condition in real Banach spaces. The results obtained in this paper are more general than many previous results and complement them.