A Continuous Analogue of Lattice Path Enumeration

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.

The Risks of Discretization: What Is Lost in (Even Good) Levels-of-Automation Schemes

In this reaction to David Kaber’s article in this volume, the author points to an inherent problem in applying any “levels” scheme to the continuous, multidimensional space of human–automation relationships and behaviors. Discretization inherently carves a continuous, analog space into discrete blocks that, the claim is, one can treat homogenously. The author provides a counterexample using a common automated e-mail filtering system as an example of how applying a single “level-of-automation” category to the whole system (or even to information-processing stages of components within it) misrepresents and suppresses details about what the system is actually doing and how it interacts with human users. Discretization can be highly productive if it pares away confusing detail that distracts from underlying explanatory relationships, but, the author argues, not enough is known about human–automation interaction in all its variability to effectively suppress detail. Thus one needs the better models Kaber is calling for before being able to create an effective levels-of-automation scheme, not vice versa.

Continuous Analog of Accelerated OS-EM Algorithm for Computed Tomography

The maximum-likelihood expectation-maximization (ML-EM) algorithm is used for an iterative image reconstruction (IIR) method and performs well with respect to the inverse problem as cross-entropy minimization in computed tomography. For accelerating the convergence rate of the ML-EM, the ordered-subsets expectation-maximization (OS-EM) with a power factor is effective. In this paper, we propose a continuous analog to the power-based accelerated OS-EM algorithm. The continuous-time image reconstruction (CIR) system is described by nonlinear differential equations with piecewise smooth vector fields by a cyclic switching process. A numerical discretization of the differential equation by using the geometric multiplicative first-order expansion of the nonlinear vector field leads to an exact equivalent iterative formula of the power-based OS-EM. The convergence of nonnegatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem for consistent inverse problems. We illustrate through numerical experiments that the convergence characteristics of the continuous system have the highest quality compared with that of discretization methods. We clarify how important the discretization method approximates the solution of the CIR to design a better IIR method.

Based ARMK10 Electromagnetic Sensor A/D Converter Design and Implementation

This paper-based design Kinetics ARM-Cortex M4 MK10DN512ZLL10 microprocessor detecting through the use of electromagnetic sensors has 10mH 100mA alternating current conductor, a continuous analog signal obtained through the solenoid to the LM358 chip amplifier many times the signal amplifier circuit , and then the analog signal is converted by the algorithm into a digital signal, and these values ​​can be obtained through a filtering process after a smooth and accurate digital microprocessor for processing.

NUMERICAL INVESTIGATION OF BIFURCATIONS AND TRANSITIONS OF JOSEPHSON VORTICES IN INHOMOGENEOUS JUNCTIONS

We consider in-line and overlap geometry models of Josephson junctions with point or rectangular inhomogeneity and investigate the effect of their location on the Josephson vortices and the current. We analyze numerically the critical dependencies "current-magnetic field" caused by one- and two-point current injections. The obtained results elucidate the relation between these critical curves and the fractions of the injection current at the ends of the junction. We also find out similarities between the exponentially shaped junctions, and those with inhomogeneity at the end when a two-point current injection is present. We juxtapose the critical curves of the distinct junctions with inner inhomogeneity and discuss the similarity between them and the Josephson junctions with phase shifts. The transitions of Josephson junctions from a superconducting mode to a resistive one as bifurcations of the static solutions of appropriately posed multiparametric compound boundary- and eigenvalue problems are interpreted and solved using the continuous analog of Newton method.