Indefinite Case Research Articles

Indefinite Nilsolitons and Einstein Solvmanifolds

A nilsoliton is a nilpotent Lie algebra $\mathfrak{g}$ with a metric such that $\operatorname{Ric}=\lambda \operatorname{Id}+D$, with $D$ a derivation. For indefinite metrics, this determines four different geometries, according to whether $\lambda$ and $D$ are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that $D$ must satisfy when it is nonzero. For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold of the form $\mathfrak{g}\rtimes \mathbb{R}^k$. Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form $\mathfrak{g}\rtimes \mathbb{R}^k$ that determine a nilsoliton metric on $\mathfrak{g}$ by restriction. We show with examples that, unlike in the Riemannian case, one cannot establish a correspondence between the full classes of Einstein solvmanifolds and nilsolitons.

On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

Quantum algorithms for solving the Quantum Linear System (QLS) problem are among the most investigated quantum algorithms of recent times, with potential applications including the solution of computationally intractable differential equations and speed-ups in machine learning. A fundamental parameter governing the efficiency of QLS solvers is κ, the condition number of the coefficient matrix A, as it has been known since the inception of the QLS problem that for worst-case instances the runtime scales at least linearly in κ [Harrow, Hassidim and Lloyd, PRL 103, 150502 (2009)]. However, for the case of positive-definite matrices classical algorithms can solve linear systems with a runtime scaling as κ, a quadratic improvement compared to the the indefinite case. It is then natural to ask whether QLS solvers may hold an analogous improvement. In this work we answer the question in the negative, showing that solving a QLS entails a runtime linear in κ also when A is positive definite. We then identify broad classes of positive-definite QLS where this lower bound can be circumvented and present two new quantum algorithms featuring a quadratic speed-up in κ: the first is based on efficiently implementing a matrix-block-encoding of A−1, the second constructs a decomposition of the form A=LL† to precondition the system. These methods are widely applicable and both allow to efficiently solve BQP-complete problems.

Infinite horizon multiobjective optimal control of stochastic cooperative linear-quadratic dynamic difference games

This article is concerned with the infinite horizon stochastic cooperative linear-quadratic (LQ) dynamic difference game in both the regular and the indefinite cases. Firstly, due to the constraints imposed on the weighting matrices and the linearity of the dynamic system, the costs are shown to be convex spontaneously for the regular stochastic cooperative LQ difference game, which yields the equivalence between the minimization of the weighted sum of costs and the Pareto optimal control. Secondly, the Pareto optimal control is derived for the regular game on the ground of the solution to the weighted algebraic Riccati equation (WARE) under exact observability, and then Pareto solutions are identified via the optimal feedback gain matrices and the solution to the weighted algebraic Lyapunov equation (WALE). Moreover, a new criterion which is also necessary and sufficient is developed to guarantee the costs to be convex for the indefinite case, and the Pareto optimality is investigated based on the solutions to the weighted generalized algebraic Riccati equation (WGARE) and the weighted generalized algebraic Lyapunov equation (WGALE) combining with the semidefinite programming (SDP). Finally, the fishery management game in the economy is presented to illustrate the obtained results.

On a class number formula of Hurwitz

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.

Ground states and multiple solutions for Choquard-Pekar equations with indefinite potential and general nonlinearity

This paper focuses on the study of ground states and multiple solutions for the following non-autonomous Choquard-Pekar equation:{−Δu+V(x)u=(W⁎F(x,u))f(x,u),x∈RN(N≥2),u∈H1(RN), where V∈C(RN,R). We consider first the case V changes sign which turns the problem into a indefinite case, and obtain the existence of nontrivial solution and infinitely many distinct pairs of solutions under a local super-linear condition assumed on the nonlinearity. For the case V is 1-periodic and positive, ground state solution and infinitely many solutions are established further by using the generalized Nehari manifold method. We finally give some non-existence criteria via a generalized Pohožaev identity established for the general potentials V and W.

GROUND STATES FOR A FRACTIONAL REACTION-DIFFUSION SYSTEM

In this paper, we prove the existence of the ground state of a strongly indefinite fractional reaction-diffusion system based on the Non-Nehari method established by Tang-Chen-Lin-Yu [J. Differ. Equ., 2020(268), 4663-4690]. In particular, neither any monotonicity condition nor any Ambrosetti-Rabinowitz growth condition is required. To our knowledge, this is the first result about the ground states with the strongly indefinite case for fractional reaction-diffusion system.

Periodic motion for FPU lattice dynamical systems with the strongly indefinite case

Sun and Ma (2013, 2019) proved the existence of nonzero T-periodic solution for the Fermi–Pasta–Ulam (FPU) lattice dynamical system. In this paper, we prove the existence of ground state solutions, i.e., the least energy of nonzero T-periodic solution for FPU lattice dynamical systems via Non-Nehari method developed recently by Tang et al. (2020) without strict monotonous condition.

Indefinite mean-field type linear–quadratic stochastic optimal control problems

This paper focuses on indefinite stochastic mean-field linear–quadratic (MF-LQ, for short) optimal control problems, which allow the weighting matrices for state and control in the cost functional to be indefinite. The solvability of stochastic Hamiltonian system and Riccati equations is presented under indefinite case. The optimal controls in open-loop form and closed-loop form are derived, respectively. In particular, dynamic mean–variance portfolio selection problem can be formulated as an indefinite MF-LQ problem to tackle directly. Another example also sheds light on the theoretical results established.

Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization

We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary points which satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a recently introduced approach involving the use of the numeral called Grossone. In particular, recalling that in principle the CG method is well posed only when solving positive definite linear systems, our proposal exploits the use of grossone to enhance the performance of the CG, allowing the computation of negative curvature directions in the indefinite case, too. Our overall method could be used to significantly generalize the theory in state-of-the-art literature. Moreover, it straightforwardly allows the solution of Newton’s equation in optimization frameworks, even in nonconvex problems. We remark that our iterative procedure to compute a negative curvature direction does not require the storage of any matrix, simply needing to store a couple of vectors. This definitely represents an advance with respect to current results in the literature.

Stabilization control for Itô stochastic system with indefinite state and control weight costs

In standard linear-quadratic (LQ) control, the first step in investigating infinite-horizon optimal control is to derive the stabilisation condition with the optimal LQ controller. This paper focuses on the stabilisation of an Itô stochastic system with indefinite control and state-weighting matrices in the cost functional. A generalised algebraic Riccati equation (GARE) is obtained via the convergence of the generalised differential Riccati equation (GDRE) in the finite-horizon case. More importantly, the necessary and sufficient stabilisation conditions for indefinite stochastic control are obtained. One of the key techniques is that the solution of the GARE is decomposed into a positive semi-definite matrix that satisfies the singular algebraic Riccati equation (SARE) and a constant matrix that is an element of the set satisfying certain linear matrix inequality conditions. Using the equivalence between the GARE and SARE, we reduce the stabilisation of the general indefinite case to that of the definite case, in which the stabilisation is studied using a Lyapunov functional defined by the optimal cost functional subject to the SARE.

Application of Mobile Technology in Community Transmission Surveillance for 2020 Corona Virus Disease

The corona virus disease (COVID-19) outbreak reached community transmission scenario in Nigeria less than 2 months from its first case on 27thFebruary 2020; a scenario which requires aggressive and effective contact tracing, identification, monitoring and reporting of disease trend. Surveillance is one of the working strategies in responding to community spread. Currently the COVID-19 community spread response mechanism relies on contact tracing and community testing, requiring a huge testing capacity, and its coverage is bound to be undermined by the lack of accessibility to the typical rural Nigerians. Massive testing is currently threatened by the short supply of test kits reported globally, inadequate trained manpower and limited molecular laboratories and testing centers, which cause delay in sample certification. This paints a picture of a massive unreached population in our communities and a predictable escalation of community transmission due to non- or untimely identification, isolation and management of cases. These gaps could be closed by the application of an appropriate tool that engages remote and real-time functions into the process. This article documents the feasibility of employing the mobile telephone opportunities for community transmission monitoring and case reporting of 2020 COVID-19 pandemic, with crowdsourcing and whistle-blowing effects; as an effective alternative to indefinite single case recording and unidirectional contact tracing.

Pareto optimality in the infinite horizon cooperative difference game

This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary conditions for the existence of the Pareto solutions are put forward. Furthermore, two conditions are presented to guarantee that zero does not belong to the Lagrange multiplier set. In addition, it is shown that the necessary conditions are also sufficient under certain convexity assumptions and a transversality condition. Next, the indefinite linear quadratic case is discussed. For a fixed initial state, under the condition of controllability, the necessary conditions are put forward. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional as well as a transversality condition provide the sufficient conditions for a control to be Pareto optimal. For an arbitrary initial state, if the system is stabilisable, then the solvability of the related algebraic Riccati equation provides a sufficient condition under which all Pareto optimal strategies are obtained by the weighted sum minimisation method.

Monotonicity-Based Reconstruction of Extreme Inclusions in Electrical Impedance Tomography

The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions, as well as to the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.

Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems

Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems.

An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.

Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations

We discuss the stochastic linear-quadratic (LQ) optimal controlproblem with Poisson processes under the indefinite case. Based on thewellposedness of the LQ problem, the main idea is expressed by thedefinition of relax compensator that extends the stochasticHamiltonian system and stochastic Riccati equation with Poissonprocesses (SREP) from the positive definite case to the indefinite case. Wemainly study the existence and uniqueness of the solution for thestochastic Hamiltonian system and obtain the optimal control withopen-loop form. Then, we further investigate the existence anduniqueness of the solution for SREP in some special case and obtainthe optimal control in close-loop form.

Uniqueness and stability in triple porosity thermoelasticity

A general model for a triple porosity thermoelastic solid is presented in the linear anisotropic case. This allows for cross coupling of inertia coefficients, and cross coupling of interaction coefficients representing actions between pressures in the macro, meso and micro structures. Sufficient conditions are derived to demonstrate uniqueness and stability when the elastic coefficients are, in a precise sense, positive. Uniqueness is further demonstrated in the dynamical problem when the elastic coefficients are not sign-definite and possess only the major symmetry. An indication is given as to how one would proceed to obtain continuous dependence upon the inital data in the Hölder sense. The proof of uniqueness in the indefinite elasticity tensor case involves a logarithmic convexity method which proceeds by a novel choice of functional.

Homoclinics for strongly indefinite almost periodic second order Hamiltonian systems

AbstractUnder certain assumptions, we prove the existence of homoclinic solutions for almost periodic second order Hamiltonian systems in the strongly indefinite case. The proof relies on a careful analysis of the energy functional restricted to the generalized Nehari manifold, and the existence and fine properties of special Palais–Smale sequences.

Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear differential equationu″+cu′+(a+(t)−μa−(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill's equation. We prove the existence of 2m−1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

Implications of dissipativity on stability of economic model predictive control—The indefinite linear quadratic case

In contrast to the conventional model predictive control (MPC) approach to control of a given system where a positive–definite objective function is employed, economic MPC employs a generic cost which is related to the ‘economics’ of the process as the objective function in the regulation layer. Often, stability proofs of the closed-loop system are based on strict-dissipativity of the system with respect to this objective function. In this paper, we focus on linear systems with indefinite quadratic costs. We show that while strict–dissipativity guarantees stability of the closed–loop system, it is not required. Hence we formulate a necessary and sufficient condition that guarantees asymptotic stability of the closed loop system. This condition comes down to the existence of two distinct storage functions for which the system is dissipative.