Separation Theorem Research Articles

Comparative index and Lidskii angles for symplectic matrices

In this paper we establish a connection between two important concepts from the matrix analysis, which have fundamental applications in the oscillation theory of differential equations. These are the traditional Lidskii angles for symplectic matrices and the recently introduced comparative index for a pair of Lagrangian planes. We show that the comparative index can be calculated by a specific argument function of symplectic orthogonal matrices, which are constructed from the Lagrangian planes. The proof is based on a topological property of the symplectic group and on the Sturmian separation theorem for completely controllable linear Hamiltonian systems. We apply the main result in order to present elegant proofs of certain important properties of the comparative index.

On an Extension of Hoffmann’s Separation Theorem for Quadratic Forms

AbstractLet $p$ and $q$ be anisotropic non-degenerate quadratic forms of dimension $\geq 2$ over an arbitrary field $F$, let $s$ be the unique non-negative integer for which $2^s<{\textrm{dim}( p)} \leq 2^{s+1}$, and let $k$ be the dimension of the anisotropic part of $q$ after extension to $F(p)$. A recent conjecture of the author then asserts that ${\textrm{dim}( q)}$ must lie within $k$ of an integer multiple of $2^{s+1}$. This statement, which holds trivially if $k \geq 2^s -1$, represents a natural generalization of the well-known separation theorem of Hoffmann, bridging a gap between it and certain classical results on the Witt kernels of function fields of quadrics. In the present article, we prove the conjecture in the case where $\textrm{char}(F) \neq 2$ and ${\textrm{dim}( p)}> 2k - 2^{s-1}$. This implies, in particular, that the conjecture holds if $\textrm{char}(F) \neq 2$ and either $k \leq 2^{s-1} + 2^{s-2}$ or ${\textrm{dim}( p)} \geq 2^s + 2^{s-1} - 4$.

Scalarization and Optimality Conditions for the Approximate Solutions to Vector Variational Inequalities in Banach Spaces

In this paper, we present the notion of approximate solutions for vector variational inequalities in Banach spaces, which extends the existing approximate solutions. It is shown that, under the cone subconvexlikeness, our new approximate solutions can be characterized by the approximate solutions of linear scalar problem by means of the convex separation theorem. For the nonconvex cases, based on the nonconvex separation theorem with a special scalar functional introduced by Hiriart–Urruty, we get the nonlinear scalarization results for the approximate solutions. In order to establish optimality conditions without convexity assumption, we calculate the subdifferential of the Hiriart–Urruty nonlinear scalar functional. And based on the scalar characterizations, optimality conditions are obtained by using Ekeland variational principle and Fermat rule for scalar optimization problems. Finally, we consider the special case of vector inequality problems in finite dimensional Euclidean space, and establish some relations between the approximate solutions of vector inequality problems and nonsmooth vector optimization problems.

Convex MPC for exclusion constraints

This article develops model predictive control for exclusion constraints with a priori guaranteed strong system theoretic properties, which is implementable via computationally highly efficient, strictly convex quadratic programming. The proposed approach deploys safe tubes in order to ensure intrinsically nonconvex exclusion constraints via closed polyhedral constraints. A safe tube is constructed by utilizing the separation theorem for convex sets, and it is practically obtained from the solution of a strictly convex quadratic programming problem. A safe tube is deployed to efficiently optimize a predicted finite horizon control process via another strictly convex quadratic programming problem.

The Fisher Separation Theorem And Capital Budgeting Decisions Of Quoted Firms In Nigeria

This study explored the Fisher separation theorem and capital budgeting decisions of quoted firms on the Nigerian stock exchange. A sample of 60 questionnaires were filled and returned by staffs from particular sectors like manufacturing, health and agriculture. Descriptive statistics was employed to illustrate the data while the Spearman rank order correlation test was used to determine if a significant relationship exist among the variables. From the estimates, the Net Present Value and Modified Internal Rate of Return are regularly employed by firms in making capital budgeting decisions. Also, when firms employ capital budgeting tools, it creates wealth for both managers and shareholders, providing support for the Fisher’s separation theorem. Finally, a correlation coefficient of 0.827 reflect a positive and linear relationship between capital budgeting decision and Shareholders’ value creation. Thus, an increase in capital budgeting decisions result to an increase in value creation. This outcome is consistent with findings from other economies and previous studies. It thus recommends that firms in the Nigerian environment should ensure they play down on shareholders’ desire for dividends and instead redirect their funds to more investments by employing suitable capital budgeting decisions.

General stochastic separation theorems with optimal bounds

Phenomenon of stochastic separability was revealed and used in machine learning to correct errors of Artificial Intelligence (AI) systems and analyze AI instabilities. In high-dimensional datasets under broad assumptions each point can be separated from the rest of the set by simple and robust Fisher’s discriminant (is Fisher separable). Errors or clusters of errors can be separated from the rest of the data. The ability to correct an AI system also opens up the possibility of an attack on it, and the high dimensionality induces vulnerabilities caused by the same stochastic separability that holds the keys to understanding the fundamentals of robustness and adaptivity in high-dimensional data-driven AI. To manage errors and analyze vulnerabilities, the stochastic separation theorems should evaluate the probability that the dataset will be Fisher separable in given dimensionality and for a given class of distributions. Explicit and optimal estimates of these separation probabilities are required, and this problem is solved in the present work. The general stochastic separation theorems with optimal probability estimates are obtained for important classes of distributions: log-concave distribution, their convex combinations and product distributions. The standard i.i.d. assumption was significantly relaxed. These theorems and estimates can be used both for correction of high-dimensional data driven AI systems and for analysis of their vulnerabilities. The third area of application is the emergence of memories in ensembles of neurons, the phenomena of grandmother’s cells and sparse coding in the brain, and explanation of unexpected effectiveness of small neural ensembles in high-dimensional brain.

Blessing of dimensionality at the edge and geometry of few-shot learning

In this paper we present theory and algorithms enabling classes of Artificial Intelligence (AI) systems to continuously and incrementally improve with a priori quantifiable guarantees – or more specifically remove classification errors – over time. This is distinct from state-of-the-art machine learning, AI, and software approaches. The theory enables building few-shot AI correction algorithms and provides conditions justifying their successful application. Another feature of this approach is that, in the supervised setting, the computational complexity of training is linear in the number of training samples. At the time of classification, the computational complexity is bounded by few inner product calculations. Moreover, the implementation is shown to be very scalable. This makes it viable for deployment in applications where computational power and memory are limited, such as embedded environments. It enables the possibility for fast on-line optimisation using improved training samples. The approach is based on the concentration of measure effects and stochastic separation theorems and is illustrated with an example on the identification faulty processes in Computer Numerical Control (CNC) milling and with a case study on adaptive removal of false positives in an industrial video surveillance and analytics system.

Output event-triggered tracking synchronization of heterogeneous systems on directed digraph via model-free reinforcement learning

In this study, an event-triggered controller for the output tracking synchronization problem in heterogeneous multi-agent systems on a directed digraph is developed. Initially, the design of the distributed observer is proposed to estimate the information of the leader. Owing to the communication among agents, there exists a coupling behavior in the design of controllers for followers. The separation theorem is exploited to realize decoupling between agents. Then, an optimal conventional controller and event-triggered controller are designed to achieve the synchronization service of the output tracking problem for agents; the latter also improves the efficiency precisely by deliberately aperiodic sampling. Additionally, we prove that the inter-event time interval is strictly positive in the event-triggered controller; that is, the Zeno behavior can be avoided. Furthermore, the optimal control gain is obtained in a model-free manner. More specifically, a model-free reinforcement learning algorithm is investigated to learn the control gain, which does not require the followers’ dynamics. Finally, two simulation examples show the availability of the algorithm.

A converse of Sturm's separation theorem

We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a turning point in the principal part of the equation. Related results are discussed.

Modelling Orebody Structures: Block Merging Algorithms and Block Model Spatial Restructuring Strategies Given Mesh Surfaces of Geological Boundaries

This paper describes a framework for capturing geological structures in a 3D block model and improving its spatial fidelity given new mesh surfaces. Using surfaces that represent geological boundaries, the objectives are to identify areas where refinement is needed, increase spatial resolution to minimize surface approximation error, reduce redundancy to increase the compactness of the model and identify the geological domain on a block-by-block basis. These objectives are fulfilled by four system components which perform block-surface overlap detection, spatial structure decomposition, sub-blocks consolidation and block tagging, respectively. The main contributions are a coordinate-ascent merging algorithm and a flexible architecture for updating the spatial structure of a block model when given multiple surfaces, which emphasizes the ability to selectively retain or modify previously assigned block labels. The techniques employed include block-surface intersection analysis based on the separable axis theorem and ray-tracing for establishing the location of blocks relative to surfaces. To demonstrate the robustness and applicability of the proposed block merging strategy in a more narrow setting, it is used to reduce block fragmentation in an existing model where surfaces are not given and the minimum block size is fixed. To obtain further insight, a systematic comparison with octree subblocking subsequently illustrates the inherent constraints of dyadic hierarchical decomposition and the importance of inter-scale merging. The results show the proposed method produces merged blocks with less extreme aspect ratios and is highly amenable to parallel processing. The overall framework is applicable to orebody modelling given geological boundaries, and 3D segmentation more generally, where there is a need to delineate spatial regions using mesh surfaces within a block model.

Distribution and number of focal points for linear Hamiltonian systems

In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by Reid (1971) about the existence of conjoined bases with an invertible first component.

Linear and Fisher Separability of Random Points in the d-Dimensional Spherical Layer and Inside the d-Dimensional Cube.

Stochastic separation theorems play important roles in high-dimensional data analysis and machine learning. It turns out that in high dimensional space, any point of a random set of points can be separated from other points by a hyperplane with high probability, even if the number of points is exponential in terms of dimensions. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining the intrinsic dimensionality of data and for explaining various natural intelligence phenomena. In this paper, we refine the estimations for the number of points and for the probability in stochastic separation theorems, thereby strengthening some results obtained earlier. We propose the boundaries for linear and Fisher separability, when the points are drawn randomly, independently and uniformly from a d-dimensional spherical layer and from the cube. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.

Fisher-Modigliani-Miller organisational finance theory and the financialisation of contemporary societies

Apart from followers as Milton Friedman, Paul Samuelson, Ronald Coase, and Maurice Allais, most economists abandoned Irving Fisher’s economic framework after the post-1929 Great Crisis. Without citing Fisher however, in 1958 Franco Modigliani and Merton Miller reutilised his framework to found the organisational finance theory awarded several Nobel prizes. This organisational finance theory is highly supportive of the financialisation of societies. It advocates that organisations exist to maximise their owners’/shareholders’ wealth, a ‘perfect markets fallacy’, and the elimination of taxes and regulations. Furthermore, it contends an unverified separation theorem between financial and productive sectors, or among financial, investing, and operating decisions.

Optimality results for a specific fractional problem

<p style='text-indent:20px;'>In this paper, one minimizes a fractional function over a compact set. Using an exact separation theorem, one gives necessary optimality conditions for strict optimal solutions in terms of Fréchet subdifferentials. All data are assumed locally Lipschitz.</p>

Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

Abstract We prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.

Models of set theory in which the separation theorem fails

We use a finite-support product of Jensen-minimal forcings to define a model of set theory in which the separation theorem fails for the projective classes and , for a given .

Separation Theorem for Average Optimal Control for Hybrid Systems of Variable Dimension

We consider the problem of average optimal control for a linear hybrid system whose continuous motion alternates with discrete changes (switchings) that change the state space. The initial system state is random. The control quality is characterized by the mean value of a quadratic functional. Switching times and their number are not known in advance. They are determined by minimizing the functional. For the problem under consideration, the classical separation principle does not hold. We prove the so-called conditional separation principle. We also show sample applications of conditional and classical separation principles.

Optimality Conditions for Strictly Efficient Solutions in Set-valued Optimization

A new kind of tangent derivative, M-derivative, for set-valued function is introduced with help of a modified Dubovitskij-Miljutin cone. Several generalized pseudoconvex set-valued functions are introduced. When both the objective function and constraint function are M-derivative, under the assumption of near conesubconvexlikeness, by applying properties of the set of strictly efficient points and a separation theorem for convex sets, Fritz John and Kuhn-Tucker necessary optimality conditions are obtained for a point pair to be a strictly efficient element of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn-Tucker sufficient optimality condition is obtained for a point pair to be a strictly efficient element of set-valued optimization problem.

$$\delta $$-ideals of a pseudocomplemented semilattice

In this paper we study $$\delta $$ -ideals of a pseudocomplemented semilattice. We give some characterizations of $$\delta $$ -ideals. We also prove Stone type separation theorem for $$\delta $$ -ideals of a pseudocomplemented semilattice. We show that the set of $$\delta $$ -ideals of a pseudocomplemented semilattice form a distributive pseudocomplemented lattice. We also prove that the set of principal $$\delta $$ -ideals form a Boolean algebra.

Optimality Conditions and Scalarization of Approximate Quasi Weak Efficient Solutions for Vector Equilibrium Problem

This paper is devoted to the investigation of optimality conditions for approximate quasi weak efficient solutions for a class of vector equilibrium problem (VEP). First, a necessary optimality condition for approximate quasi weak efficient solutions to VEP is established by utilizing the separation theorem with respect to the quasirelative interior of convex sets and the properties of the Clarke subdifferential. Second, the concept of approximate pseudoconvex function is introduced and its existence is verified by a concrete example. Under the assumption of introduced convexity, a sufficient optimality condition for VEP in sense of approximate quasi weak efficiency is also presented. Finally, by using Tammer’s function and the directed distance function, the scalarization theorems of the approximate quasi weak efficient solutions of the VEP are proposed.