Guest Column

Algebraic Natural Proofs is a recent framework which formalizes the type of reasoning used for proving most lower bounds on algebraic computational models. This concept is similar to and inspired by the famous natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower bounds, but, unlike in the boolean case, it is an open problem whether this constitutes a barrier for proving super-polynomial lower bounds for strong models of algebraic computation. From an algebraic-geometric viewpoint, it is also related to basic questions in Geometric Complexity Theory (GCT), and from a meta-complexity theoretic viewpoint, it can be seen as an algebraic version of the MCSP problem. We survey the recent work around this concept which provides some evidence both for and against the existence of an algebraic natural proofs barrier, with an emphasis on the di erent viewpoints and the connections to other areas.

The Effects of Absorbing Materials on the Homogeneity of Composite Heating by Microwave Radiation

When cured in a microwave, flat thin composite panels can experience even heat distribution throughout the laminate. However, as load and geometric complexity increase, the electromagnetic field and resulting heat distribution is altered, making it difficult to cure the composite homogeneously. Materials that absorb and/or reflect incident electromagnetic radiation have the potential to influence how the field behaves, and therefore to tailor and improve the uniformity of heat distribution. In this study, an absorber was applied to a composite with non-uniform geometry to increase heating in the location which had previously been the coldest position, transforming it into the hottest. Although this result overshot the desired outcome of temperature uniformity, it shows the potential of absorbing materials to radically change the temperature distribution, demonstrating that with better regulation of the absorbing effect, a uniform temperature distribution is possible even in non-uniform composite geometries.

Program Complex for Automated Calculation of the Geometry of Impact Units of Machines

Improving the designs of impact mining machines in order to increase the productivity of drilling operations requires calculations of the geometric parameters of impact units. The greatest effect when the impact is applied to the rock is given by an impact pulse corresponding to the resistance forces of the object being destroyed. In turn, the shape and parameters of the impact pulse are determined by the geometry of the colliding bodies. Analytical methods for analyzing dynamic processes in impact systems involve the use of a very complex mathematical apparatus, which does not allow us to quickly solve the problems of engineering design of machines and mechanisms. The authors of this article have developed a numerical method for calculating and analyzing impact pulses generated in the machine system by bodies of any geometric complexity. The reliability of the theoretical approaches is confirmed by the results of a physical experiment. The developed software allows you to quickly and accurately solve the problem of finding and justifying rational geometric parameters of impact nodes of machines.

Geometric Complexity and the Information-Theoretic Comparison of Functional-Response Models

The assessment of relative model performance using information criteria like AIC and BIC has become routine among functional-response studies, reflecting trends in the broader ecological literature. Such information criteria allow comparison across diverse models because they penalize each model's fit by its parametric complexity—in terms of their number of free parameters—which allows simpler models to outperform similarly fitting models of higher parametric complexity. However, criteria like AIC and BIC do not consider an additional form of model complexity, referred to as geometric complexity, which relates specifically to the mathematical form of the model. Models of equivalent parametric complexity can differ in their geometric complexity and thereby in their ability to flexibly fit data. Here we use the Fisher Information Approximation to compare, explain, and contextualize how geometric complexity varies across a large compilation of single-prey functional-response models—including prey-, ratio-, and predator-dependent formulations—reflecting varying apparent degrees and forms of non-linearity. Because a model's geometric complexity varies with the data's underlying experimental design, we also sought to determine which designs are best at leveling the playing field among functional-response models. Our analyses illustrate (1) the large differences in geometric complexity that exist among functional-response models, (2) there is no experimental design that can minimize these differences across all models, and (3) even the qualitative nature by which some models are more or less flexible than others is reversed by changes in experimental design. Failure to appreciate model flexibility in the empirical evaluation of functional-response models may therefore lead to biased inferences for predator–prey ecology, particularly at low experimental sample sizes where its impact is strongest. We conclude by discussing the statistical and epistemological challenges that model flexibility poses for the study of functional responses as it relates to the attainment of biological truth and predictive ability.

Geometric complexity and the information-theoretic comparison of functional-response models

The assessment of relative model performance using information criteria like AIC and BIC has become routine among functional-response studies, reflecting trends in the broader ecological literature. Such information criteria allow comparison across diverse models because they penalize each model's fit by its parametric complexity --- in terms of their number of free parameters --- which allows simpler models to outperform similarly fitting models of higher parametric complexity. However, criteria like AIC and BIC do not consider an additional form of model complexity, referred to as geometric complexity, which relates specifically to the mathematical form of the model. Models of equivalent parametric complexity can differ in their geometric complexity and thereby in their ability to flexibly fit data. Here we use the Fisher Information Approximation criterion to compare, explain, and contextualize how geometric complexity varies across a large compilation of single-prey functional-response models --- including prey-, ratio-, and predator-dependent formulations --- reflecting varying levels of phenomenological generality and varying apparent degrees and forms of non-linearity. Because a model's geometric complexity varies with the data's underlying experimental design, we also sought to determine which designs are best at leveling the playing field among functional-response models. Our analyses illustrate (1) the large differences in geometric complexity that exist among functional-response models, (2) there is no experimental design that can minimize these differences across all models, and (3) even the qualitative nature by which some models are more or less flexible than others is reversed by changes in experimental design. Failure to appreciate geometric complexity in the empirical evaluation of functional-response models may therefore lead to biased inferences for predator--prey ecology, particularly at low experimental sample sizes where the relative effects of geometric complexity are strongest. We conclude by discussing the statistical and epistemological challenges that geometric complexity poses for the study of functional responses as it relates to the attainment of biological truth and predictive ability.

A data-driven method to predict achievability of clinical objectives in IMRT

When specifying a clinical objective for a target volume and normal organs/tissues in IMRT planning, the user may not be sure if the defined clinical objective could be achieved by the optimizer. To this end, we propose a novel method to predict the achievability of clinical objectives upfront before invoking the optimization. A new metric called “Geometric Complexity (GC)” is used to estimate the achievability of clinical objectives. Essentially GC is the measure of the number of “unmodulated” beamlets or rays that intersect the Region-of-interest (ROI) and the target volume. We first compute the geometric complexity ratio (GCratio) between the GC of a ROI in a reference plan and the GC of the same ROI in a given plan. The GCratio of a ROI indicates the relative geometric complexity of the ROI as compared to the same ROI in the reference plan. Hence GCratio can be used to predict if a defined clinical objective associated with the ROI can be met by the optimizer for a given case. We have evaluated the proposed method on six Head and Neck cases using Pinnacle3 (version 9.10.0) Treatment Planning System (TPS). Out of total of 42 clinical objectives from six cases accounted in the study, 37 were in agreement with the prediction, which implies an agreement of about 88% between predicted and obtained results. The results indicate the feasibility of using the proposed method in head and neck cases for predicting the achievability of clinical objectives.

FabHandWear

Current hand wearables have limited customizability, they are loose-fit to an individual's hand and lack comfort. The main barrier in customizing hand wearables is the geometric complexity and size variation in hands. Moreover, there are different functions that the users can be looking for; some may only want to detect hand's motion or orientation; others may be interested in tracking their vital signs. Current wearables usually fit multiple functions and are designed for a universal user with none or limited customization. There are no specialized tools that facilitate the creation of customized hand wearables for varying hand sizes and provide different functionalities. We envision an emerging generation of customizable hand wearables that supports hand differences and promotes hand exploration with additional functionality. We introduce FabHandWear, a novel system that allows end-to-end design and fabrication of customized functional self-contained hand wearables. FabHandWear is designed to work with off-the-shelf electronics, with the ability to connect them automatically and generate a printable pattern for fabrication. We validate our system by using illustrative applications, a durability test, and an empirical user evaluation. Overall, FabHandWear offers the freedom to create customized, functional, and manufacturable hand wearables.

Influence of Build Orientation, Geometry and Artificial Saliva Aging on the Mechanical Properties of 3D Printed Poly(ε-caprolactone)

Additive manufacturing of polymers has evolved from rapid prototyping to the production of functional components/parts with applications in distinct areas, ranging from health to aeronautics. The possibility of producing complex customized geometries with less environmental impact is one of the critical factors that leveraged the exponential growth of this processing technology. Among the several processing parameters that influence the properties of the parts, the geometry (shape factor) is amid less reported. Considering the geometric complexity of the mouth, including the uniqueness of each teething, this study can contribute to a better understanding of the performance of polymeric devices used in the oral environment for preventive, restorative, and regenerative therapies. Thus, this work aims to evaluate 3D printed poly(ε-caprolactone) mechanical properties with different build orientations and geometries. Longitudinal and transversal toolpaths produced specimens with parallelepiped and tubular geometry. Moreover, as it is intended to develop devices for dentistry, the influence of artificial saliva on mechanical properties was determined. The research concluded that the best mechanical properties are obtained for parallelepiped geometry with a longitudinal impression and that aging in artificial saliva negatively influences all the mechanical properties evaluated in this study.

Fractal Dimension as Quantifier of EEG Activity in Driving Simulation

Dynamical systems and fractal theory methodologies have been proved useful for the modeling and analysis of experimental datasets and, in particular, for electroencephalographic signals. The computation of the fractal dimension of approximation curves in the plane enables the assignment of numerical values to bioelectric recordings in order to discriminate between different states of the observed system. The procedure does not require the stationarity of the signals nor extremely long segments of data. In previous works, we checked that this parameter is a good index for brain activity. In this paper, we consider this measurement in order to quantify the geometric complexity of the brain waves in states of rest and during vehicle driving simulation in different scenarios. This work presents evidence that the fractal dimension allows the detection of the brain bioelectric changes produced in the areas that carry out the different driving simulation tasks, increasing with their complexity.