Minimal Sufficient Conditions Research Articles

On the invariant factors and the minimal polynomial assignments of linear systems

In this note, the invariant factors assignment and the minimal polynomial assignment problems for multiple-input linear systems are studied. It is found that for any invariant factors with the geometry multiplicity of each eigenvalue less than or equal to the number of the inputs, the invariant factors assignment problems are solvable if and only if the controllability indices set of the linear system satisfies certain condition. For the minimal polynomial assignment problem, a necessary and sufficient condition on the solvability is proposed. Three numerical examples are provided to illustrate the proposed results.

Spelling out leptonic CP violation in the language of invariant theory

In terms of flavor invariants, we establish the intimate connection between leptonic CP violation in the canonical seesaw model for neutrino masses and that in the seesaw effective field theory (SEFT). For the first time, we calculate the Hilbert series and explicitly construct the primary flavor invariants in the SEFT by considering both the dimension-five Weinberg operator ${\cal O}^{\alpha \beta}_5 = \overline{\ell^{}_{\alpha \rm L}} \widetilde{H} \widetilde{H}^{\rm T} \ell^{\rm C}_{\beta \rm L}$ and the dimension-six operator ${\cal O}^{\alpha \beta}_6 = \overline{\ell^{}_{\alpha \rm L}} \widetilde{H} {\rm i} \not\!\partial \widetilde{H}^\dagger \ell^{}_{\beta \rm L}$ at the tree-level matching. The inclusion of only the Wilson coefficients $C^{\alpha \beta}_5$ and $C^{\alpha \beta}_6$ already enables the SEFT to incorporate all physical information about the full seesaw model. Moreover, the minimal sufficient and necessary conditions for CP conservation both in the SEFT and in the full theory are clarified, and the matching between the flavor invariants in both theories is accomplished. Through the matching of flavor invariants, the CP asymmetries necessary for successful leptogenesis are directly linked to those in neutrino-neutrino and neutrino-antineutrino oscillations at low energies. Surprisingly, it is revealed that the precise measurements of $C^{\alpha \beta}_5$ and $C^{\alpha \beta}_6$ in low-energy experiments are powerful enough to probe the full seesaw model, including CP violation for cosmological matter-antimatter asymmetry.

Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.</p></abstract>

Stochastic functional Kolmogorov equations, I: Persistence

This work (Part (I)) together with its companion (Part (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) is devoted to extinction. The main techniques used in this paper include the newly developed functional Itô formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined covering, improving, and substantially extending the existing literature. Furthermore, our results reduce to that in the existing literature of Kolmogorov systems when there is no past dependence.

New trends in modeling and control of hybrid systems

New trends in modeling and control of hybrid systems

Definability of Local Abelian Groups by Their Splitting Rings

This paper is devoted to the problem (Glaz, Vinsonhaler, Wickless): When a $$p$$ -local torsion-free group is determined by the collection of its splitting rings? For $$p$$ -local torsion-free groups with quadratic and cubic minimal splitting fields necessary and sufficient condition has been found according to which $$p$$ -local group $$G$$ is determined by the collection of its splitting rings.

Event-Triggered-Based Adaptive Sliding Mode Control for T–S Fuzzy Systems With Actuator Failures and Signal Quantization

This article investigates the problem of event-triggered-based sliding mode control (SMC) for a class of Takagi-Sugeno (T-S) fuzzy systems with actuator failures and signal quantization. The classical dynamic uniform quantization strategy is employed to quantize data in both sensor-controller side and controller-actuator side. A new event-triggered adaptive SMC scheme is proposed to stabilize the fault closed-loop systems, where the event-triggered condition is based on quantized state vectors. Reachability of the proposed sliding surface can be ensured by the designed control scheme. Furthermore, the existence of minimal inter-event time and sufficient conditions under which zeno behavior can be avoided is analyzed and presented. Finally, two examples are provided to demonstrate the effectiveness of the proposed new design techniques.

Minimal Sufficient Conditions for Structural Observability/Controllability of Composite Networks via Kronecker Product

In this article, we consider composite networks formed from the Kronecker product of smaller networks. We find the observability and controllability properties of the product network from those of its constituent smaller networks. The overall network is modeled as a Linear-Structure-Invariant (LSI) dynamical system where the underlying matrices have a fixed zero/non-zero structure but the non-zero elements are potentially time-varying. This approach allows to model the system parameters as free variables whose values may only be known within a certain tolerance. We particularly look for minimal sufficient conditions 1 1 We emphasize that a minimal sufficient condition is not necessarily a necessary and sufficient condition. In fact, it implies that among all sufficient conditions that may result in an event, this condition is the least conservative but usually is not necessary; see [1] for details. on the observability and controllability of the composite network, which have a direct application in distributed estimation and in the design of networked control systems. The methodology in this article is based on the structured systems analysis and graph theory, and therefore, the results are generic, i.e., they apply to almost all non-zero choices of free parameters. We show the controllability/observability results for composite product networks resulting from full structural-rank systems and self-damped networks. We provide an illustrative example of estimation based on Kalman filtering over a composite network to verify our results.

Minimal condition for shortest vectors in lattices of low dimension

For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG+, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.

Scalarizations for a set optimization problem using generalized oriented distance function

The aim of this paper is to establish scalarizations for minimal and weak minimal solutions of a set optimization problem using generalized oriented distance function introduced by Crespi et al. (Math Methods Oper Res 63:87–106, 2006). The solution concepts are based on a partial set order relation on the family of nonempty bounded sets proposed by Karaman et al. (Positivity 22:783–802, 2018). Finally, we also provide existence results for minimal solutions and sufficient conditions for the solution sets to be closed.

A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

Predictive characterization of mixtures of Markov chains

Predictive constructions are a powerful way of characterizing the probability laws of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps in designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process $\mathbf{X}$. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.

Minimal sufficient statistical experiments on von Neumann algebras

A statistical experiment on a von Neumann algebra is a parametrized family of normal states on the algebra. This paper introduces the concept of minimal sufficiency for statistical experiments in such operator algebraic situations. We define equivalence relations of statistical experiments indexed by a common parameter set by completely positive or Schwarz coarse-graining and show that any statistical experiment is equivalent to a minimal sufficient statistical experiment unique up to normal isomorphism of outcome algebras. We also establish the relationship between the minimal sufficiency condition for a statistical experiment in this paper and those for subalgebra. These concepts and results are applied to the concatenation relation for completely positive channels with general input and outcome von Neumann algebras. In the case of the quantum-classical channel corresponding to the positive-operator valued measure (POVM), we prove the equivalence of the minimal sufficient condition previously proposed by the author and that in this paper. We also give a characterization of the discreteness of a POVM up to postprocessing equivalence in terms of the corresponding quantum-classical channel.

Proof of covering minimality by generalizing the notion of independence

A method is proposed for obtaining lower bounds for the length of the shortest cover and complexity of the minimal cover based on the notion of independent family of sets. For the problem of minimization of Boolean functions, we provide the functions and construct coverings by faces of the set of unit vertices for which the suggested lower bounds can be achieved in the case of five or more variables. The lower bounds, based on independent sets, are unreachable and cannot be used as sufficient conditions for minimality of such functions.

Learning by stimulation avoidance: A principle to control spiking neural networks dynamics.

Learning based on networks of real neurons, and learning based on biologically inspired models of neural networks, have yet to find general learning rules leading to widespread applications. In this paper, we argue for the existence of a principle allowing to steer the dynamics of a biologically inspired neural network. Using carefully timed external stimulation, the network can be driven towards a desired dynamical state. We term this principle “Learning by Stimulation Avoidance” (LSA). We demonstrate through simulation that the minimal sufficient conditions leading to LSA in artificial networks are also sufficient to reproduce learning results similar to those obtained in biological neurons by Shahaf and Marom, and in addition explains synaptic pruning. We examined the underlying mechanism by simulating a small network of 3 neurons, then scaled it up to a hundred neurons. We show that LSA has a higher explanatory power than existing hypotheses about the response of biological neural networks to external simulation, and can be used as a learning rule for an embodied application: learning of wall avoidance by a simulated robot. In other works, reinforcement learning with spiking networks can be obtained through global reward signals akin simulating the dopamine system; we believe that this is the first project demonstrating sensory-motor learning with random spiking networks through Hebbian learning relying on environmental conditions without a separate reward system.

Minimal conditions for implications of Gronwall–Bellman type

Gronwall–Bellman type inequalities entail the following implication: if a sufficiently integrable function satisfies a certain homogeneous linear integral inequality, then it is nonpositive. We present a minimal (necessary and sufficient) condition on the Borel measure underlying the integrals for this implication to hold. The condition is also a necessary prerequisite for any nontrivial bound on solutions to inhomogeneous linear integral inequalities of Gronwall–Bellman type.

Geometric constrained variational calculus. III: The second variation (Part II)

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

Geometric constrained variational calculus. II: The second variation (Part I)

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

An axiomatic integral and a multivariate mean value theorem

In order to investigate minimal sufficient conditions for an abstract integral to belong to the convex hull of the integrand, we propose a system of axioms under which it happens. If the integrand is a continuous $\mathbf {R}^{n}$ -valued function over a path-connected topological space, we prove that any such integral can be represented as a convex combination of values of the integrand in at most n points, which yields an ultimate multivariate mean value theorem.

Negative Imaginary Lemmas for Descriptor Systems

This technical note studies the negative imaginary properties of descriptor linear systems based on state-space realizations. Under the assumption of a minimal realization, necessary and sufficient conditions are established to characterize the negative imaginary properties of descriptor systems in terms of linear matrix inequalities with equality constraints. In particular, a negative imaginary lemma, a strict negative imaginary lemma and a lossless negative imaginary lemma are developed. A multiple-input and multiple-output RLC circuit network is used as an illustrative example to validate the developed theory.