Separation Theorem Research Articles

Judicial Independence vs Parliamentary Supremacy: Reignition of debate vis-à-vis Israel and India

Huge demonstrations are going on in State of Israel against the Judicial reforms being proposed by the PM Netanyahu led Government.1 In India also war of words is ongoing between senior government functionaries and Higher Judiciary over the collegium process and its impact. Per se the healthy discussion and negotiation between different organs of the government is not bad but it should not be with an aim to reduce their importance. Judicial independence is the hallmark of Rule of Law in any democracy.2 It only protects individuals’ liberty but also keeps a check on the state excesses or biasness. Indisputably, the parliament has an exclusive right of law making but the moot question remain whether power to amend includes power to destroy. There must be checks and balances on all the three organs of the government. Even Judiciary cannot be allowed free run without any accountability and reforms must keep going but any Judicial reforms cannot be allowed to strike at the root of Judicial independence itself as it would destroy the institution. In this paper I have tried to explain the current situation in Israel where the proposed reforms being brought in the name of Judicial accountability and Judicial reform threaten the very concept of judicial independence. It strikes at the very basic theorem of separation of Power and would promote autocracy and dictatorship in the long run. As Israel is a parliamentary democracy, the executive heads already influence the decisions of the Knesset in a major way & if they get to have the same in Judicial appointment then the political leaders ruling the State of Israel will have all the powers with very less accountability. In comparison the situation in India is different. While verbal arguments are going on between Judiciary and Executive it is more or less limited to Judicial appointment with others spheres of Judicial independence like Judicial Review not part of the public debate. The paper also argues about the need for Judicial independence in a democracy and the Power of Parliament to amend is not absolute and unlimited.

Distributed secure consensus control of nonlinear multi-agent systems under sensor and actuator attacks

In this paper, we focus on an output secure consensus control issue for nonlinear multi-agent systems (MASs) under sensor and actuator attacks. Followers in an MAS are in strict-feedback form with unknown control directions and unknown dead-zone input, where both sensors and nonlinear characteristics of dead-zone in actuators are paralyzed by malicious attacks. To deal with sensor attacks, uncertain dynamics in individual follower are separated by a separation theorem, and estimation parameters are introduced for compensating and mitigating the influence from adversaries. The influence from actuator attacks are treated as a total displacement in a dead-zone nonlinearity, and an upper bound, as well as its estimation, is introduced for this displacement. The dead-zone nonlinearity, sensor attacks and unknown control gains are gathered together regarded as composite unknown control directions, and Nussbaum functions are utilized to address the issue of unknown control directions. A distributed secure consensus control strategy is thus developed recursively for each follower in the framework of surface control method. Theoretically, the stability of the closed-loop MAS is analyzed, and it is proved that the MAS achieves output consensus in spite of nonlinear dynamics and malicious attacks. Finally, theoretical results are verified via a numerical example and a group of electromechanical systems.

Exact separation theorem for disjoint closed sets in Hilbert spaces

Exact separation theorem for disjoint closed sets in Hilbert spaces

Hedging and the regret theory of the firm

This paper examines the production and hedging decisions of the competitive firm under price uncertainty when the firm is not only risk averse but also regret averse. Regret-averse preferences are characterized by a modified utility function that includes disutility from having chosen ex post suboptimal alternatives. The extent of regret depends on the difference between the actual profit and the maximum profit attained by making the optimal production, and hedging decisions had the firm observed the true realization of the random output price. While the separation theorem holds under regret aversion, the prevalence of hedging opportunities may have perverse effect on the firm’s optimal output level, particularly when the firm is sufficiently regret averse. The full-hedging theorem, however, does not hold. We derive sufficient conditions under which the regret-averse firm’s optimal futures position is an under-hedge (over-hedge). We further show that the firm optimally increases (decreases) its futures position when the price risk possesses more positive (negative) skewness.

A new discrete theory of pseudoconvexity

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.

Some Algebraic and Topological Structures of Laplace Transformable Functions

Abstract: In this work, the set of all functions that are Laplace transformable with regard to their structures both algebraic and topological, is taken into account. Certain topological properties of the set of Laplace transformable functions with the help of a metric are described. Also, we determine the proofs of the statements that the set of all Laplace transformable functions is a commutative semi-group with respect to the convolution operation as well as an Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Laplace transformable functions is defined and the proof that the Laplace transformable functions' space is complete with our metric is given. The separability theorem and that the Laplace transformable functions' space is disconnected are also discussed. Keywords: Abelian group, Commutative semi-group, Disconnected space, Laplace transform, Separability theorem.

Disturbance Rejection and Control System Design Based on an Improved Equivalent-Input-Disturbance Approach

This article presents an improved equivalent-input-disturbance (EID) approach to actively rejecting exogenous disturbances for a plant. A control system based on the approach includes a new EID estimator constructed by embedding integrals to the conventional EID estimator. These integrals enable the estimator to improve the disturbance-estimation precision without amplifying the effect of measurement noise. Stability conditions of the control system are obtained using separation theorem. The system design ensures that the control system satisfies the stability conditions. Experiments of a rotational control system demonstrate the validity of the approach.

Operational Absolutely Optimal Dynamic Control of the Stochastic Differential Plant’s State by Its Output

The problem of synthesizing the average-optimal control law for a dynamic plant subject to random disturbances, if its state variables are measured partially or with random errors, is considered. Using the method of a posteriori sufficient coordinates (SCs), the complexity of constructing the well-known interval-optimal Mortensen controller is described and a much simpler algorithm for finding its operational-optimal analog is obtained. The new controller does not require the solution of the corresponding Bellman equation in inverse time, since it is optimal in the sense of a time-varying criterion. This makes it possible to disregard information about the future behavior of the object and reduces the procedure for finding the dependence of a control on sufficient coordinates to direct-time integration of the Fokker–Planck–Kolmogorov equation and to solving a problem of parametric nonlinear programming. The application of the obtained algorithm is demonstrated by the example of a linear-quadratic-Gaussian problem, as a result of which a new operational version of the well-known separation theorem is formulated. It represents a stochastic control device as a combination of a linear Kalman–Bucy filter and a linear operational-optimal positional controller. The latter differs from the traditional interval-optimal controller by the well-known gain and does not require the solution of the corresponding matrix Riccati equation in inverse time

The Theoretical Substantiation of Cost Accounting in the Light of Conflicting Approaches

Desde hace algunos años se viene sosteniendo en Alemania una discusión acerca de la fundamentación teórica de la contabilidad de costes, así como de la necesidad de mantener un sistema de costes independiente. El abanico de planteamientos posibles abarca desde la total integración de la contabilidad de costes en la contabilidad externa hasta el de una separación lo más completa posible de ambos sistema contables. Esta aportación pretende presentar, desde el punto de vista alemán, cómo se ha desarrollado la discusión y cuál es la situación actual. Como conclusión del trabajo se aboga por una separación, sustentada teóricamente, entre ambos sistemas de contabilidad. El trabajo persigue varios objetivos: (a) identificación de las raíces históricas de las bases teóricas de la contabilidad de costes, (b) presentación de la problemática de la empresa y de su estructuración, (c) definición de los planteamientos de separación y de sus teoremas, (d) análisis de las perspectivas de investigación y de su fundamentación teórica, (e) orientación metodológica de la contabilidad de costes dentro del sistema de planificación y dirección, (f) presentación y evaluación de las recientes contribuciones acerca de los fundamentos teóricos, (g) fundamento teórico de un sistema independiente de contabilidad de costes.

Reliable and Secure Short-Packet Communications in Untrusted Diamond Relay Networks

This paper investigates short-packet communications over a diamond relay network with two untrusted relays (potential eavesdroppers) with the purpose of guaranteeing reliability and security simultaneously. As the performance metric, reliable-and-secure probability (RSP), considering both reliability and secrecy, is defined from the perspective of physical layer security. An analytical approach is proposed to investigate reliability and security by taking into account the characteristics of the short-packet transmission. RSPs are numerically obtained via the analytical framework by utilizing the source-channel separation theorem for the source-to-relay transmission and chief executive officer (CEO) problem analyses for the relay-to-destination transmission. It is found that the optimal RSP is achieved when the contributions from the source-relay transmission and the relay-destination transmission are balanced, even without support from a friendly jamming signal or artificial noise.

Time-consistent lifetime portfolio selection under smooth ambiguity

<p style='text-indent:20px;'>This paper studies the optimal consumption, life insurance and investment problem for an income earner with uncertain lifetime under smooth ambiguity model. We assume that risky assets have unknown market prices that result in ambiguity. The individual forms his belief, that is, the distribution of market prices, according to available information. His ambiguity attitude, which is similar to the risk attitude described by utility function <inline-formula><tex-math id="M1">\begin{document}$ U $\end{document}</tex-math></inline-formula>, is represented by an ambiguity preference function <inline-formula><tex-math id="M2">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. Under the smooth ambiguity model, the problem becomes time-inconsistent. We derive the extended Hamilton-Jacobi-Bellman (HJB) equation for the equilibrium value function and equilibrium strategy. Then, we obtain the explicit solution for the equilibrium strategy when both <inline-formula><tex-math id="M3">\begin{document}$ U $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> are power functions. We find that a more risk- or ambiguity-averse individual will consume less, buy more life insurance and invest less. Moreover, we find that the Tobin-Markowitz separation theorem is no longer applicable when ambiguity attitude is taken into consideration. The investment strategy will change with the characteristics of the decision maker, such as risk attitude, ambiguity attitude and age.</p>

Stochastic Separation Theorems: How Geometry May Help to Correct AI Errors

Stochastic Separation Theorems: How Geometry May Help to Correct AI Errors

Neural joint source-channel coding via Bernoulli latent straight-through estimator

Under infinite block length, from Shannon's separation theorem, it is well-known that by independent design of source coding and channel coding, the optimal throughput — the channel capacity — can be reached with careful design of the corresponding functionalities. However, when restricted to finite block length, the separation theorem does not necessarily hold, and hence joint source-channel coding (JSCC) theorem has been raised as an alternative strategy for achieving the capacity of the channel. However, as JSCC formulates highly non-convex problem which cannot be directly solved analytically, recently, deep learning has been proposed as a key enabler for JSCC. While most work on deep-learning-based JSCC focused on transmission of continuous signals through wireless channels, we focus in this paper the case of information-theoretic channel, namely binary symmetric channel, which can give useful insights on information-theoretic perspective on JSCC. Unlike recent work on deep-learning-based, or neural, JSCC for discrete channels that considered score function estimator to train the JSCC, in this paper, we improve the performance of neural JSCC by estimating the gradient more precisely as compared to the previous approach. Key idea is to consider soft codeword during training to enable path-wise gradient estimator which is proven to have lower variance than the score-function estimator. Experimental results on MNIST and CIFAR-10 datasets show that the proposed neural JSCC outperforms the previous work on JSCC for discrete channels, validating the effectiveness of the proposed gradient computation technique.

Separation theorems in the commutative algebra of -rings and applications

In this paper we state and prove ad hoc “Separation Theorems” of the so-called Smooth Commutative Algebra, the Commutative Algebra of -rings. These results are formally similar to the ones we find in (ordinary) Commutative Algebra. However, their proofs are not so straightforward, since they depend on the introduction of the key concept of “smooth saturation.” As an application of these theorems we present an interesting result that sheds light on the natural connection between the smooth Zariski spectrum and smooth real spectrum of a -ring, the -analog of the real spectrum of a commutative unital ring.

Group Invariant Operators and Some Applications to Norm-Attaining Theory

In this paper, we study geometric properties of the set of group invariant continuous linear operators between Banach spaces. In particular, we present group invariant versions of the Hahn–Banach separation theorems and elementary properties of the invariant operators. This allows us to contextualize our main applications in the theory of norm-attaining operators; we establish group invariant versions of the properties \(\alpha \) of Schachermayer and \(\beta \) of Lindenstrauss, and present relevant results from this theory in this (much wider) setting. In particular, we generalize Bourgain’s result, which says that if X has the Radon–Nikodým property, then X has the G-Bishop–Phelps property for G-invariant operators whenever \(G \subseteq \mathcal {L}(X)\) is a compact group of isometries on X.

Unified theory of thermodynamics and stochastic thermodynamics for nonlinear Langevin systems driven by non-conservative forces

We construct a unified theory of thermodynamics and stochastic thermodynamics for classical nonequilibrium systems driven by non-conservative forces, using the recently developed covariant Ito-Langevin theory. The thermodynamic forces are split into a conservative part and a non-conservative part. Thermodynamic functions are defined using the reference conservative system. Work and heat are partitioned into excess parts and house-keeping parts, which are due to, respectively, conservative forces and non-conservative forces. Excess entropy production (EP) and house-keeping EP are analogously defined. The splitting of thermodynamic forces is subjected to an arbitrariness resembling a gauge symmetry, with each gauge defining a reference conservative Langevin system. In the special Gibbs gauge, the nonequilibrium steady state (NESS) is characterized by Gibbs canonical distribution, the excess heat agrees with that defined by Hatano and Sasa, and the excess EP agrees with that of Glansdorff and Prigogine, i.e., it is the time rate of the second-order variation of system entropy near the NESS. Adopting the Gibbs gauge, and focusing on the excess parts of thermodynamic quantities, a complete analogy between thermodynamics of non-conservative systems and that of conservative systems is established. One important consequence of this analogy is that both the free energy and excess EP are minimized at NESS. Our theory therefore constitutes a statistical foundation both for the steady-state thermodynamics theory due to Sasa and Tasaki and for the stability theory of NESS due to Glansdorff and Prigogine. These results are valid even if the system is far from equilibrium. By studying detailed fluctuation theorem, we find striking differences between systems with symmetric kinetic matrices and those with asymmetric kinetic matrices. For systems with asymmetric kinetic matrices, the total EP is the sum of house-keeping EP, excess EP, and pumped entropy. Entropy pumping is an exchange of entropy between the system and environment without necessarily involving dissipation. In the presence of entropy pumping, the system may behave as either a demon or an antidemon. Fluctuation theorems and work relations are derived both for total work and for excess work. For systems with symmetric kinetic matrices, there is no entropy pumping, yet in the Gibbs gauge, the excess work and house-keeping work each satisfies a separate fluctuation theorem. We illustrate our theory using many concrete examples.

Infinitely many virtual geometric triangulations

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.

Global Adaptive Control for Uncertain Nonlinear Systems under Non-Lipschitz Condition with Quantized States

A novel adaptive control method is designed for the uncertain nonlinear systems with quantized states under the condition that system nonlinearities do not satisfy the Lipschitz condition. In this paper, the global control for an uncertain nonlinear system with quantized states in the case that the system nonlinearities do not satisfy the Lipschitz condition is first achieved. Separation theorem is used to model the uncertainties appropriately and then an adaptive term is designed to help attenuate the effects of system uncertainties. Furthermore, the global boundedness of all signals of the closed-loop system is proved based on the Lyapunov stability theorem. Finally, simulation results are given to demonstrate the effectiveness of the proposed methods.

Comparative Study of Markov Chain Filtering Schemas for Stabilization of Stochastic Systems under Incomplete Information

The object under investigation is a controllable linear stochastic differential system affected by some external statistically uncertain piecewise continuous disturbances. They are directly unobservable but assumed to be a continuous-time Markov chain. The problem is to stabilize the system output concerning a quadratic optimality criterion. As is known, the separation theorem holds for the system. The goal of the paper is performance analysis of various numerical schemes applied to the filtering of the external Markov input for system stabilization purposes. The paper briefly presents the theoretical solution to the considered problem of optimal stabilization for systems with the Markov jump external disturbances: the conditions providing the separation theorem, the equations of optimal control, and the ones defining the Wonham filter. It also contains a complex of the stable numerical approximations of the filter, designed for the time-discretized observations, along with their accuracy characteristics. The approximations of orders 12, 1, and 2 along with the classical Euler–Maruyama scheme are chosen for the comparison of the Wonham filter numerical realization. The filtering estimates are used in the practical stabilization of the various linear systems of the second order. The numerical experiments confirm the significant influence of the filtering precision on the stabilization performance and superiority of the proposed stable schemes of numerical filtering.

Mean–variance efficient large portfolios: a simple machine learning heuristic technique based on the two-fund separation theorem

Mean–variance efficient large portfolios: a simple machine learning heuristic technique based on the two-fund separation theorem