Nonpositive Elements Research Articles

Solving Linear Programming Problems by Reducing to the Form with an Obvious Answer

The article considers a method for solving a linear programming problem (LPP), which requires finding the minimum or maximum of a linear functional on a set of non-negative solutions of a system of linear algebraic equations with the same unknowns. The method is obtained by improving the classical simplex method, which when involving geometric considerations, in fact, generalizes the Gauss complete exclusion method for solving systems of equations. The proposed method, as well as the method of complete exceptions, proceeds from purely algebraic considerations. It consists of converting the entire LPP, including the objective function, into an equivalent problem with an obvious answer. For the convenience of converting the target functional, the equations are written as linear functionals on the left side and zeros on the right one. From the coefficients of the mentioned functionals, a matrix is formed, which is called the LPP matrix. The zero row of the matrix is the coefficients of the target functional, $a_{00}$ is its free member. The algorithms are described and justified in terms of the transformation of this matrix. In calculations the matrix is a calculation table. The method under consideration by analogy with the simplex method consists of three stages. At the first stage the LPP matrix is reduced to a special 1-canonical form. With such matrices one of the basic solutions of the system is obvious, and the target functional on it is $ a_{00}$, which is very convenient. At the second stage the resulting matrix is transformed into a similar matrix with non-positive elements of the zero column (except $a_{00}$), which entails the non-negativity of the basic solution. At the third stage the matrix is transformed into a matrix that provides non-negativity and optimality of the basic solution. For the second stage the analog of which in the simplex method uses an artificial basis and is the most time-consuming, two variants without artificial variables are given. When describing the first of them, along the way, a very easy-to-understand and remember analogue of the famous Farkas lemma is obtained. The other option is quite simple to use, but its full justification is difficult and will be separately published.

What Does the Operator Algebra of Quantum Statistics Tell Us about the Objective Causes of Observable Effects?

Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a subjective lack of information regarding the physical reality of the system. In the present paper, I explore how the operator formalism accommodates the vast number of possible states and measurements by characterizing its essential function as a description of causality relations between initial conditions and subsequent observations. It is shown that any complete description of causality must involve non-positive statistical elements that cannot be associated with any directly observable effects. The necessity of non-positive elements is demonstrated by the uniquely defined mathematical description of ideal correlations which explains the physics of maximally entangled states, quantum teleportation and quantum cloning. The operator formalism thus modifies the concept of causality by providing a universally valid description of deterministic relations between initial states and subsequent observations that cannot be expressed in terms of directly observable measurement outcomes. Instead, the identifiable elements of causality are necessarily non-positive and hence unobservable. The validity of the operator algebra therefore indicates that a consistent explanation of the various uncertainty limited phenomena associated with physical objects is only possible if we learn to accept the fact that the elements of causality cannot be reconciled with a continuation of observable reality in the physical object.

Risk measuring under model uncertainty

The framework of this paper is that of risk measuring under uncertainty which is when no reference probability measure is given. To every regular convex risk measure on $\mathcal{C}_{b}(\Omega)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless nonpositive elements of $\mathcal{C}_{b}(\Omega)$. We prove that the convex risk measure has a dual representation with a countable set of probability measures absolutely continuous with respect to a certain probability measure in this class. To get these results we study the topological properties of the dual of the Banach space L1(c) associated to a capacity c. As application, we obtain that every G-expectation $\mathbb{E}$ has a representation with a countable set of probability measures absolutely continuous with respect to a probability measure P such that P(|f|) = 0 if and only iff $\mathbb{E}(|f|)=0$. We also apply our results to the case of uncertain volatility.

On the Bauer-Muir transformation for continued fractions and its applications

We show that the Bauer-Muir transformation is useful also for continued fractions with non-positive elements. Some examples of applications of this transformation are studied.

Nonpositive elements in whitehead groups

If G is a group, then ZG is its integral group ring, GL(ZG) is the direct limit l& GL,(ZG), Ki(ZG) is the abelianization of GL(ZG) and Wh(G) is the quotient Ki(ZG)/z. These algebraic definitions are motivated by topology. If K is a finite CW-complex which deformation retracts to a subcomplex L, write K-+L. For such a pair (K, L), there is an associated element T(K, L) of Wh(sriL) which determines the simple homotopy type of K relative to L. M.M. Cohen has defined the dimension of an element 7. of Wh(rriL) to be:

Duality theorems for linear systems and convex systems

We show that, if ( F → u X) is a linear system, Ω ⊂ X a convex target set and h: X → R ̄ a convex functional, then, under suitable assumptions, the computation of inf h({y ϵ F ¦ u(y) ϵ Ω}) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of u ∗(X ∗) , or of the infima on F of Lagrangians, involving elements of u ∗(X ∗) . Also, we prove similar results for a convex system ( F → u X) and the convex cone Ω of all non-positive elements in X.

The local ergodic theorem and semigroups of nonpositive operators

Letting ( T( t): t ⩾ 0) be a strongly continuous semigroup of contraction operators on L 1(X , M ,λ) we consider the question of the almost-everywhere convergence of ( 1 α ) ∝ 0 α T(t)ƒ dt as α → 0 + (the local ergodic theorem). The limit is known to exist in the case where the semigroup is positive. In this paper we consider the question for general semigroups which may contain nonpositive elements. We show that a maximal ergodic inequality is equivalent to the local ergodic theorem. We also present some extensions to the local ergodic theorem. All the results of this paper also hold for N-parameter semigroups.