Improved convergence results for the Nelder-Mead simplex method in low dimensional spaces

We prove two results on the convergence of the Nelder-Mead simplex method. Both theorems prove the convergence of the simplex vertices to a common limit point. The first result is an improved variant of the convergence theorems of [7] and [8], while the second one proves the convergence with probability 1.

Johanna Seibt’s Process Ontology of Categorical Inference: On Nomological Axiomatics and Category Projection

Drawing from a Sellarsian realist-naturalist epistemology, we trace different levels of cognitive hierarchy procedures through which a representational system learns to update its own states and improve its ‘map-making’ capabilities from pre-conscious operations which modulate base-localization functions, to patterns of epistemic revision and integration at the conceptual and theoretical levels, producing a nomological double of its world. We show how ontological theorization becomes diachronically coordinated with and constrained by empirical science, and how the formal-quantitative kernel of scientific theories corresponds to qualitative-conceptual determinations at the structural level. Following Johanna Seibt’s characterization of ontology as a theory of categorial inference, we trace the preservation of inferential semantic structure across ontological theories in relation to model languages and provide provisional indications to coordinate Seibt’s account with a convergent realist assessment of systematic modeling, defining the epistemological conditions for articulating the preservation of formal structure in theories toward a limit-point of enquiry.

Richardson-Kalitkin method in abstract description

An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem Ρ{{\Rho}} is considered, the solution of which results in a real number uu. To solve this problem, a numerical method is used, that is, the set Hℝ{H\subset \mathbb{R}} and the mapping uh:Hℝ{u_h:H\to\mathbb{R}} are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set HH and uh{u_h} can be expanded in a convergent series in powers of h:uh=u+c1hk+...{h:u_h=u+c_1h^k+...}. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for uu and cc from two values of uh{u_h}. The question of using a larger number of uh{u_h} values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.

Energetics of collapsible channel flow with a nonlinear fluid-beam model

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.

Nonlinear Buckling of Fixed Functionally Graded Material Arches Under a Locally Uniformly Distributed Radial Load

Functionally graded material (FGM) arches may be subjected to a locally radial load and have different material distributions leading to different nonlinear in-plane buckling behavior. Little studies is presented about effects of the type of material distributions on the nonlinear in-plane buckling of FGM arches under a locally radial load in the literature insofar. This paper focuses on investigating the nonlinear in-plane buckling behavior of fixed FGM arches under a locally uniformly distributed radial load and incorporating effects of the type of material distributions. New theoretical solutions for the limit point buckling load and bifurcation buckling loads and nonlinear equilibrium path of the fixed FGM arches under a locally uniformly distributed radial load that are subjected to three different types of material distributions are derived. The comparisons between theoretical and ANSYS results indicate that the theoretical solutions are accurate. In addition, the critical modified geometric slendernesses of FGM arches related to the switches of buckling modes are also derived. It is found that the type of material distributions of the fixed FGM arches affects the limit point buckling loads and bifurcation buckling loads as well as the nonlinear equilibrium path significantly. It is also found that the limit point buckling load and bifurcation buckling load increase with an increase of the modified geometric slenderness, the localized parameter and the proportional coefficient of homogeneous ceramic layer as well as a decrease of the power-law index p of material distributions of the FGM arches.

CONDITIONS ON UNIQUENESS OF LIMIT POINT AND COMPLETENESS IN CONE POLYGONAL METRIC SPACES

This paper discusses cone polygonal metric spaces. We analyze some characteristics derived from convergence and Cauchyness of sequences. Our result consists of some conditions on uniqueness of limit point and completeness in cone polygonal metric spaces.

Fractional linear maps in general relativity and quantum mechanics

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

Multiple Steady States in the Photocatalytic Reactor for Colored Compounds Degradation

The paper reports the occurrence of multiple steady-state zones in most of the constructions of fixed-bed photocatalytic reactors. Such a phenomenon has not been ever observed in a field of photocatalytic reactors. The simulation has been provided for a common case in a photocatalysis—the degradation of colored compounds. The mathematical model of the photocatalytic reactor with immobilized bed has been stated by a simple ideal mixing model (analogous to the CSTR model). The solution has been continued by the two parameters—the Damköhler number and the absorption coefficient related to the inlet stream concentration. Some branches of steady states include the limit point. The performed two-parametric continuation of the limit point showed the cusp bifurcation point. Besides the numerical simulation, the physical explanation of the observed phenomenon has been provided; the multiple steady-states occurrence is controlled by light absorption–reaction rate junction. When the reaction rate is limited by the light absorption, we can say that a light barrier occurs. The dynamical simulations show that when the process is operated in a field of multiple steady states, the overall reactor efficiency is related to the reactor set-up mode.

A Relationship Between the Factor Indivisibility and the Output Elasticity of the Indivisible Factor

The paper puts forth a notion and derives a special type of production function where labour is an indivisible factor and is in the integer space. Thus, Newtonian calculus is not an appropriate method of deriving the marginal value because limit point does not exist. This shows that indivisibility determines the output elasticity. In the first part, the paper propounds a notion regarding how indivisibility determines curvature of the production function. In the second part, the paper incorporates the findings within a production function and derives a new type accordingly. Moreover, it formally derives the standard wage equation considering all the entitlements of labour, namely (a) normal wages, (b) interest and (c) rent of ability. So far, no such mathematical proof is there to support this wage composition. This paper, for the first time, derives this wage equation considering indivisibility of labour. JEL Classifications: J23, J24, J31, D24, C61, E24, L8